Reveal Geometry lessons 1-2 through 1-6 in one session: the three undefined terms, collinear and coplanar, intersections of lines and planes, segment measure and the segment addition postulate, congruent segments as an equation, length on a number line, the distance and midpoint formulas as the Pythagorean theorem and an average, and the one method — start, trip, fraction, add — behind every partitioning question on both the number line and the coordinate plane.
Subject: Honors Geometry · 62 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Lessons 1-2 through 1-6
Points, lines and planes, segments, distance, and partitioning a directed segment
Objectives
These are your teacher's stated outcomes for the week, reorganised into the order that makes each one follow from the last. Everything in the second half rests on the vocabulary in the first half, so we build in that order.
Notice that the last two outcomes appear twice, once for the number line and once for the plane. That is a hint: it is the same idea both times, and we will make sure it feels like one idea rather than two.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry chapter 1 opener
Section
Lesson 1-2
Concept
Point, line and plane are called undefined terms. That sounds like a gap in the textbook. It is the opposite — it is the textbook being careful.
Every definition explains one word using other words. If every word had to be defined, the definitions would go round in a circle forever. So geometry picks three words, describes them well enough to use, and refuses to define them. Everything else in the course is then defined from those three.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-2
Picture it
Every diagram in this chapter is a version of this picture, so read it slowly.
Figure (svg): A plane drawn as a tilted parallelogram containing a line, with three labelled collinear points on the line
The parallelogram is a lie of convenience. The plane has no edges — we draw a boundary because the page does. The same goes for the arrowheads on the line: they exist to remind you that the line does not stop.
Concept
Two adjectives that get confused constantly, largely because they sound alike and are introduced on the same page.
collinear — points that all lie on one single line
coplanar — points that all lie on one single plane
The useful fact is how easily each is satisfied. Any two points are collinear, because you can always draw the line through them. Any three points are coplanar, because you can always find a plane containing them. It only becomes an interesting question at three points for collinear and four points for coplanar.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-2
Prediction
Think about what it takes to pin something down, rather than trying to remember a rule.
Predict first
How many points do you need before asking whether they are collinear becomes a real question?
Correct: Three
Why: Through any two points there is exactly one line, so two points are always collinear — the question cannot come out false. Three is the first count where the third point might miss the line the first two determine, which makes the question worth asking.
Definition probe
Use the counting idea from the last slide rather than recall.
Sort into buckets
Is this always true, or only sometimes true?
Section
Lesson 1-2 continued
Pattern
The intersection of two figures is the set of all points they have in common. There is a pattern to what comes out, and once you see it the questions become almost automatic.
| these two figures | meet in | why |
|---|---|---|
| two distinct lines that cross | a single point | two different lines can agree in at most one place |
| a line and a plane, not lying in it | a single point | the line pierces the surface once |
| a line lying inside a plane | the whole line | every point of the line is already in the plane |
| two distinct planes that meet | a line | two flat surfaces crease along a straight fold |
| two parallel lines | nothing at all | they share no points, so the intersection is empty |
The pattern: the intersection is always simpler than the things intersecting. Planes meet in lines, lines meet in points, and points cannot get any simpler.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-2
Picture it
These two pictures cover most of what the homework will ask.
Figure (svg): On the left, two planes crossing along a shared line. On the right, a line piercing a plane at a single labelled point
When a question asks for the intersection, answer with a named object, not with a description. Say the intersection is line AB, or the intersection is point P — never the intersection is where they cross.
Check
Solve it on paper before you click.
Check your understanding
Two distinct planes intersect. What is their intersection?
Answer: B
Why: Two distinct planes either miss each other entirely or crease along a straight fold, and that fold is a line. It cannot be a single point, because if the planes share one point they must share the whole line through it in the shared direction.
Socratic
Push on the claim rather than accepting it.
Discussion prompt
Suppose two distinct planes shared exactly one point and nothing else. Why is that impossible?
Hint: What happens if you walk away from the shared point along a direction that lies in both planes?
Answer:
Take the shared point. Each plane extends flat in every direction from it, and the two planes are tilted relative to one another.
Follow the direction along which both planes run. Moving in that direction stays inside the first plane, because planes are flat and unbounded, and it stays inside the second for the same reason.
So every point along that whole direction is shared. One shared point forces an entire shared line. Sharing exactly one point would require at least one of them to curve away, and a plane cannot curve.
Section
Lesson 1-3
Concept
Three closely related objects, distinguished only by how many ends are cut off. Getting the notation right matters because the homework grades it.
| object | how far it goes | how it is written |
|---|---|---|
| line AB | forever in both directions | AB with a double-headed arrow above |
| ray AB | starts at A, forever past B | AB with a single arrow above, pointing right |
| segment AB | from A to B and stops | AB with a plain bar above |
| the measure AB | a number, not a figure | AB with nothing above it |
That last row is the one that catches people. With a bar on top it is a set of points; with nothing on top it is a length. So you can write that one segment is congruent to another, and separately that one measure equals another, but never that a segment equals a number.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-3
Notation
Read this line the way your teacher will read it when marking.
Annotate
On: \( \overline{AB} \cong \overline{CD} \quad \text{and} \quad AB = CD \)
Shorthand for the exam: bar on top means congruent, no bar means equals.
Concept
A postulate is a statement accepted without proof because it is too basic to prove from anything simpler. This one is the workhorse of the chapter.
If point B lies between points A and C, then the measure from A to B plus the measure from B to C equals the measure from A to C.
\[ AB + BC = AC \]
The word between is doing real work. It requires all three points to be collinear, with B in the middle. If B is off the line, the two short pieces add to more than the long one, and the postulate does not apply.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-3
Picture it
It is almost too obvious to state, which is exactly why it is a postulate and not a theorem.
Figure (svg): A segment from A to C with B marked between them, and three coloured bars beneath showing that AB plus BC equals AC
In problems, the postulate is usually the equation you are missing. When a question gives you three segment measures with letters in them, write this equation down before you do anything else.
Worked example
Point B lies between A and C. The measure AB is x plus 5, the measure BC is 2x minus 1, and the measure AC is 4x. Find x, and then find all three measures.
Write the segment addition postulate for these three points
Why: B is between A and C, so the two short pieces add to the long one.
\[ (x + 5) + (2x - 1) = 4x \]
Combine like terms on the left
Why: The x terms add to 3x, and the numbers add to 4.
\[ 3x + 4 = 4x \]
Solve for x
Why: Subtract 3x from both sides.
\[ x = 4 \]
Substitute back to get the three measures
Why: Put four in for x in each expression.
\[ AB = 9, \quad BC = 7, \quad AC = 16 \]
Verify: that the pieces really do add up
Why: Nine plus seven is sixteen, which is exactly AC. Had that check failed, the arithmetic went wrong somewhere and there would be no point going on.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-3
Concept
Two segments are congruent when they have the same measure. In a diagram, that is shown with matching tick marks rather than with numbers.
\[ \overline{PQ} \cong \overline{RS} \quad \text{exactly when} \quad PQ = RS \]
This is the outcome your teacher listed as applying the definition of congruent segments to find missing values. In practice it means one thing: congruence marks in a picture are an equation waiting to be written down.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-3
Picture it
The picture gives you no numbers. It gives you something better.
Figure (svg): Two segments each carrying a single tick mark, one labelled 3x minus 5 and the other x plus 7, with the solution x equals 6 beneath
One tick each means congruent, so their measures are equal. Setting 3x minus 5 equal to x plus 7 gives 2x equal to 12, so x is 6, and both segments measure 13.
Fill the middle
Segment JK carries two tick marks and so does segment LM. Their measures are 5y minus 3 and 2y plus 12.
Fill in the blanks
5y - 3 = 2y + 12 \;\Rightarrow\; 3y = 15 \;\Rightarrow\; y = 5
Why: Matching tick marks mean the segments are congruent, so their measures are equal. Subtracting 2y from both sides and adding 3 to both sides leaves 3y equal to 15, so y is 5. Checking: 5 times 5 minus 3 is 22, and 2 times 5 plus 12 is also 22.
Trap
The diagram marks segment AB as congruent to segment CD, and the student is asked to write the relationship.
Write that segment AB equals 7
Why: This puts a set of points on one side of an equals sign and a number on the other.
Write that segment AB is congruent to 7
Why: Worse: congruence is a relation between figures, and 7 is not a figure.
The diagram marks segment AB as congruent to segment CD, and the student is asked to write the relationship.
Decide which kind of object is on each side
Why: Segments are figures; measures are numbers. Match the symbol to the kind.
Write both statements correctly
Why: Segment AB is congruent to segment CD, using the congruence symbol with bars on both. The measure AB equals 7, using an equals sign with no bars at all.
Check
Solve it on paper before you click.
Check your understanding
Segment RS and segment TU are marked congruent, and each measures 12 units. Which statement is correctly written?
Answer: C
Why: A measure is a number, so the relation is equality and 12 is a legitimate right-hand side. This is the one statement where the kinds of object on both sides of the symbol actually match.
Section
Lesson 1-5
Concept
On a number line every point has one coordinate. The distance between two points is the difference of their coordinates, and because distance can never be negative, you take that difference positively.
\[ AB = \lvert b - a \rvert \]
The absolute value bars are the whole reason this formula is safe to use in either order. Subtract the smaller from the larger and you get the answer directly; subtract the other way round and the bars fix the sign for you.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-5
Picture it
One picture, and then the formula stops feeling like something to remember.
Figure (svg): A number line with point A at negative seven and point B at five, and a bar between them labelled twelve units
Five minus negative seven is twelve. You can also just count the tick marks, and on a test that counting is a genuinely good way to check an answer you got algebraically.
Error analysis
A student finds the distance between the point at 3 and the point at negative 8.
Annotate
On: \( AB = 3 - 8 = -5 \)
Two independent alarms went off here and both were ignored — the sign of the second coordinate, and the sign of the final answer. Treat a negative distance as an automatic stop-and-recheck.
Socratic
You will be told it does not matter. Make sure you know why.
Discussion prompt
Why do the expression b minus a and the expression a minus b give the same distance?
Hint: What is the relationship between b minus a and a minus b?
Answer:
Because they differ only in sign. If b minus a is 12, then a minus b is negative 12, and the two have exactly the same size.
The absolute value throws away the sign and keeps the size, so both routes land on 12.
This is also why distance is not the same thing as displacement. Distance keeps only the size; direction is information the absolute value deliberately discards. That distinction comes back the moment we start talking about directed segments.
Section
Lesson 1-6
Concept
This formula gets memorised far more often than it gets understood, which is a shame because understanding it takes about thirty seconds.
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Draw the segment between the two points. Then draw a horizontal leg and a vertical leg to make a right triangle. The horizontal leg is the difference in the x coordinates, the vertical leg is the difference in the y coordinates, and the segment you want is the hypotenuse.
OpenStax, Elementary Algebra 2e — the coordinate plane and the distance formula the distance formula
Picture it
Here is a case with friendly numbers, so the structure is visible.
Figure (svg): A coordinate plane with a segment from A at one comma two to B at four comma six, and dashed horizontal and vertical legs forming a right triangle with sides three, four and five
Run of three, rise of four, so the hypotenuse is five. Every distance-formula problem is this picture; only the numbers change, and most of the time they are not as pretty as three, four and five.
Worked example
Find the distance between the point at negative two comma one and the point at ten comma seven. Leave the answer exact, then give a decimal.
Figure (svg): A coordinate plane with the segment from negative two comma one to ten comma seven, and dashed legs of run twelve and rise six forming a right triangle
Find the run and the rise
Why: Subtract the x coordinates and then the y coordinates, in the same order both times.
\[ \Delta x = 10 - (-2) = 12, \qquad \Delta y = 7 - 1 = 6 \]
Square each and add
Why: This is the Pythagorean theorem on the two legs.
\[ 12^2 + 6^2 = 144 + 36 = 180 \]
Take the square root and simplify the radical
Why: One hundred and eighty is thirty-six times five, and thirty-six is a perfect square, so a six comes out.
\[ d = \sqrt{180} = 6\sqrt{5} \]
Give the decimal
Why: Root five is about 2.236, and six times that is about 13.4.
Verify: against the two legs
Why: The hypotenuse must be longer than either leg but shorter than the two added together. The legs are 12 and 6, so the answer must lie between 12 and 18. About 13.4 sits comfortably in that window.
OpenStax, Elementary Algebra 2e — the coordinate plane and the distance formula worked examples
Estimation
That verification step is worth building into a habit, because it catches almost every sign error for free.
Predict first
Two points are 8 apart horizontally and 15 apart vertically. Without computing, roughly what is the distance?
Correct: Between 15 and 23
Why: The hypotenuse always beats the longer leg, so the answer is more than 15, and it is always shorter than travelling along both legs, so it is less than 23. In fact it is exactly 17, which sits neatly inside that window. The bracket takes two seconds and rules out three of the four options.
Check
Solve it on paper before you click.
Check your understanding
You want the distance between the point at negative three comma four and the point at five comma negative two. Which set-up is right?
Answer: B
Why: The run is 5 minus negative 3, which is 5 plus 3, or 8. The rise is negative 2 minus 4, which is negative 6. Squaring gives 64 plus 36, or 100, and the root of that is exactly 10.
Concept
Halfway along a segment, each coordinate is halfway between the two end coordinates. Halfway between two numbers is their average, so the formula writes itself.
\[ M = \left(\dfrac{x_1 + x_2}{2}, \; \dfrac{y_1 + y_2}{2}\right) \]
Keep hold of the word average. In a moment we will want a point one third of the way along rather than halfway, and the averaging picture is what generalises. The formula, memorised as a formula, does not.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-6
Picture it
Notice how the midpoint coordinates relate to the endpoints before you read on.
Figure (svg): A coordinate plane with a segment from A at negative three comma zero to B at nine comma six, and the midpoint M marked at three comma three
Negative three and nine average to three. Zero and six average to three. So the midpoint is at three comma three, and you can see on the picture that it really does sit halfway along.
Two truths and a lie
Two of these are safe to rely on. One is a shortcut that fails.
Eliminate the wrong options
Cross out the false statement.
Survives elimination: d3
Why: Averaging two whole numbers gives a half whenever the two have opposite parity. The segment from one comma two to four comma two has midpoint at two point five comma two, and there is nothing wrong with that. Expecting whole numbers makes students distrust a perfectly correct answer.
Section
Lessons 1-5 and 1-6
Concept
Up to now a segment has had two ends and no preference between them. A directed segment has a starting point and an ending point, and the order changes the answer.
Your teacher's outcomes mention this four times: a fractional distance and a given ratio, each on a number line and on the plane. The reason it is worth four outcomes is that students who do not notice the direction get the right arithmetic and the wrong point.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-5
Picture it
Same two endpoints, same fraction, two different points.
Figure (svg): The same segment drawn twice, once with an arrow from A to B and once from B to A, with the one-third point marked in a different place each time
One third of the way from A to B is not one third of the way from B to A. Before you compute anything, underline the word after from in the question. That word tells you which endpoint is the starting point.
Pattern
This is the one idea behind all four of the partitioning outcomes, and it is worth writing on the front of your notes.
Start at the starting point. Work out the whole trip from start to finish. Take the given fraction of that trip. Add it on.
\[ P = A + k(B - A) \]
Same sentence, four situations. If you can say the sentence, you do not need four formulas.
Khan Academy, Dividing line segments in a given ratio dividing a segment in a ratio
Check
Solve it on paper before you click.
Check your understanding
A point P divides the directed segment from A to B so that the ratio of AP to PB is 3 to 5. What fraction of the way from A to B is P?
Answer: B
Why: A ratio of 3 to 5 means the whole trip is cut into 3 plus 5, which is 8 equal parts. P sits at the end of the first 3 of them, so it is three eighths of the way from A to B.
Worked example
Point A is at negative 4 and point B is at 8. Find the point one quarter of the way from A to B.
Find the whole trip
Why: Subtract the starting coordinate from the ending coordinate. This one is directed, so keep the sign.
\[ B - A = 8 - (-4) = 12 \]
Take the given fraction of the trip
Why: One quarter of twelve is three.
\[ k(B - A) = \tfrac{1}{4} \cdot 12 = 3 \]
Add the partial trip to the starting point
Why: Start at negative four and walk three units in the positive direction.
\[ P = -4 + 3 = -1 \]
Verify: by measuring both pieces
Why: From negative four to negative one is 3 units; from negative one to 8 is 9 units. Three is one quarter of the total twelve, exactly as asked, and the pieces stand in the ratio 3 to 9, which reduces to 1 to 3.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lesson 1-5
Picture it
Same segment, different question: this time the point splits the trip two parts to one part.
Figure (svg): A number line from negative six to ten with A at negative four, P at four and B at eight, and bars showing eight units then four units
Two to one means three parts in total, so k is two thirds. Two thirds of the twelve-unit trip is eight, and negative four plus eight is four. The two pieces then measure 8 and 4, which is exactly 2 to 1.
Reverse engineer
Exam questions sometimes hand you the answer and ask for the ratio. Same relationship, read the other way.
Fill in the blanks
A = 2, \; B = 20, \; P = 8 \;\Rightarrow\; AP = 6, \; PB = 12, \; AP : PB = 1 to 2
Why: From 2 to 8 is 6 units, and from 8 to 20 is 12 units. Six to twelve reduces to one to two, so P sits one third of the way from A to B. Notice that the fraction one third and the ratio one to two are the same fact in two dialects.
Worked example
Point A is at negative 4 comma negative 1, and point B is at 8 comma 5. Find the point P that partitions the directed segment from A to B in the ratio 1 to 2.
Turn the ratio into a fraction
Why: One to two means three parts in total, and P sits at the end of the first part.
\[ k = \dfrac{1}{1 + 2} = \dfrac{1}{3} \]
Find the whole trip in each direction
Why: Subtract the starting coordinates from the ending ones, keeping the signs.
\[ \Delta x = 8 - (-4) = 12, \qquad \Delta y = 5 - (-1) = 6 \]
Take one third of each
Why: The same fraction applies to the run and to the rise, because both are parts of the same trip.
\[ \tfrac{1}{3}(12) = 4, \qquad \tfrac{1}{3}(6) = 2 \]
Add each partial trip to the matching starting coordinate
Why: Negative four plus four, and negative one plus two.
\[ P = (0, \; 1) \]
Verify: by measuring both pieces
Why: From A to P the run is 4 and the rise is 2, giving a length of root twenty. From P to B the run is 8 and the rise is 4, giving root eighty, which is two root twenty. The second piece is exactly twice the first, so the ratio really is 1 to 2.
Khan Academy, Dividing line segments in a given ratio worked examples
Picture it
Seeing it once makes the arithmetic feel less arbitrary.
Figure (svg): A coordinate plane showing the segment from A at negative four comma negative one to B at eight comma five, with P marked at zero comma one, and a panel giving the run, rise and fraction
The run and rise both get cut by the same fraction, which is why the point lands on the segment rather than beside it. Cutting them by different fractions is the classic way to end up with a point that is not even on the line.
Comparison
The point of this table is how little actually changes.
Comparison matrix
| step | on a number line | on the coordinate plane |
|---|---|---|
| what a point is | one coordinate | two coordinates |
| the whole trip | the difference of the coordinates | a run and a rise, computed separately |
| take the fraction | multiply the one difference by k | multiply both the run and the rise by the same k |
| finish | add to the starting coordinate | add each to the matching starting coordinate |
| how to check | measure both pieces and compare | measure both pieces with the distance formula and compare |
One extra coordinate, one extra multiplication, one extra addition. It is genuinely the same method, which is why your teacher listed the outcomes in pairs.
Check
Solve it on paper before you click.
Check your understanding
Point A is at 1 comma 3 and point B is at 9 comma 11. Where is the point three quarters of the way from A to B?
Answer: B
Why: The run is 8 and the rise is 8. Three quarters of each is 6, and adding 6 to each starting coordinate gives 7 comma 9. Checking the pieces: from A to P is a run of 6, from P to B is a run of 2, and 6 to 2 is 3 to 1, which is exactly three quarters of the way.
Trap
Find the point one third of the way from A at 6 comma 3 to B at 12 comma 9.
Take one third of each coordinate of B
Why: That gives 4 comma 3, which the student reports as the answer.
Notice nothing is wrong
Why: Both numbers look plausible, and 4 comma 3 is not obviously absurd, so the error survives to the answer line.
Find the point one third of the way from A at 6 comma 3 to B at 12 comma 9.
Find the trip first
Why: The run is 12 minus 6, which is 6. The rise is 9 minus 3, which is also 6.
Take one third of the trip, not of the coordinates
Why: One third of 6 is 2, in each direction.
Add the partial trip to the start
Why: Six plus two is 8, and three plus two is 5, so the point is at 8 comma 5, which does lie on the segment. The wrong answer, 4 comma 3, sits outside the segment entirely — a check that catches this trap every time.
Edge cases
Definitions earn their keep at the edges, so push this one.
Discussion prompt
The formula still gives an answer if k is 2, or if k is negative. What point comes out, and does it lie on the segment?
Hint: What does walking twice the trip mean physically?
Answer:
With k equal to 2 you walk the whole trip and then the whole trip again, so you land beyond B, on the extension of the segment past the far end.
With k negative you walk backwards from A, landing on the extension past the near end, in the opposite direction from B.
Only values of k between zero and one land on the segment itself. That is a useful sanity check: if a question gives you a fraction outside that range and expects a point on the segment, something has been misread.
Missing information
Under-specified questions turn up on tests, and noticing is worth marks in itself.
Discussion prompt
Point P divides segment AB in the ratio 2 to 3. Find P, given A at 0 comma 0 and B at 10 comma 5. What is missing?
Hint: Which endpoint does the ratio start counting from?
Answer:
The direction. The question says segment AB, not the directed segment from A to B, so it has not said which endpoint is the starting point.
Starting from A gives k equal to two fifths and a point at 4 comma 2. Starting from B gives k equal to two fifths of the trip back, which puts the point at 6 comma 3. Both are legitimate readings of the words as written.
On a real test, read the phrasing carefully: directed segment from A to B settles it, and so does the ordering in the ratio AP to PB. If neither is present, state the assumption you are making at the top of your working.
Section
Consolidation
Ranking
Every one of the four partitioning outcomes is these steps in this order.
Put in order
Why: The starting point has to be settled first, because every later step measures from it. The fraction comes next because it is pure arithmetic on the ratio and does not depend on the coordinates. Then trip, then fraction of the trip, then add. The check is last and is not optional — it catches the trap of taking a fraction of the coordinates instead of the trip.
Matching
Chapter 1 hands you five tools. Questions signal which one they want.
Match the pairs
Why: The word between is always the segment addition postulate. Tick marks are always congruence, which means an equation. Halfway is the midpoint, which is the partition formula with k equal to one half. A ratio means converting to a fraction first. And a plain how far means the distance formula.
Worked example
Point M is the midpoint of the segment from A at negative 2 comma 3 to B at 6 comma 9. Find M, then find the length of the segment from A to M, then find the point one quarter of the way from A to B.
Figure (svg): A coordinate plane with the segment from A at negative two comma three to B at six comma nine, with the midpoint M and the quarter point P both marked
Find the midpoint by averaging
Why: Negative two and six average to two; three and nine average to six.
\[ M = (2, \; 6) \]
Find the length from A to M with the distance formula
Why: The run is 4 and the rise is 3, which is the friendly three-four-five triangle.
\[ AM = \sqrt{4^2 + 3^2} = \sqrt{25} = 5 \]
Find the quarter point using the partition method
Why: The full run is 8 and the full rise is 6. A quarter of each is 2 and 1.5.
\[ P = (-2 + 2, \; 3 + 1.5) = (0, \; 4.5) \]
Verify: that the three answers are consistent with one another
Why: The full length from A to B should be twice AM, so 10. Checking directly: run 8, rise 6, and the root of 64 plus 36 is the root of 100, which is 10. And the quarter point should be half way from A to M, which is the average of negative two comma three and two comma six, giving zero comma four point five. Both checks agree.
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry chapter 1 review
Explain it to yourself
If you can say this without notes, the partitioning half of the chapter is finished.
Discussion prompt
In one sentence, how do you find any point a given fraction of the way along a directed segment?
Hint: Start, trip, fraction, add.
Answer:
Start at the starting point, compute the whole trip, take that fraction of the trip, and add it on.
On a number line the trip is one number. On the plane it is a run and a rise, and the same fraction applies to both.
A ratio of m to n becomes the fraction m over m plus n, because the total number of parts is m plus n and you are stopping after m of them.
Warm-up
Close the deck. This is retrieval practice, and it is worth several times what rereading is worth.
Discussion prompt
Write down, from memory: the three undefined terms; what two distinct planes intersect in; the segment addition postulate; the distance formula; and how a ratio of m to n becomes a fraction.
Hint: Three words, one line, one postulate, one formula, one fraction.
Answer:
Point, line and plane are the three undefined terms.
Two distinct planes that meet intersect in a line.
If B is between A and C, then AB plus BC equals AC.
Distance is the square root of the run squared plus the rise squared.
A ratio of m to n gives the fraction m over m plus n.
Connect it up
No notes, twenty minutes. This is the single best revision activity for this chapter.
Draw it
Draw a number line with two labelled points, and beside it a coordinate plane with two labelled points. On each, mark the midpoint and a point one third of the way along. Beside each mark, write the calculation that produced it. Underneath, write the four-word summary: start, trip, fraction, add.
Anything you had to look up is what to review before the quiz.
Exit ticket
Be honest — this decides where the next session opens.
Predict first
Which piece of this week would you least want on a quiz tomorrow?
Correct: Whichever you picked is where we start next time.
Why: There is no wrong answer here. Naming the weak piece is worth more than another pass over the strong ones, and the chapter quiz is close enough that fixing one thing properly beats skimming all six.
Recap
Six outcomes, one week. This is the checklist your teacher will be marking against.
| you want | the formula | the check |
|---|---|---|
| length on a number line | the difference, taken positively | count the tick marks |
| distance on the plane | root of run squared plus rise squared | it must beat the longer leg |
| the midpoint | average each pair of coordinates | it must sit between the endpoints |
| a fractional distance | start plus k times the trip | k between 0 and 1 lands on the segment |
| a ratio of m to n | k equals m over m plus n | measure both pieces and compare |
McGraw Hill, Reveal Geometry, Chapter 1 — Tools of Geometry lessons 1-2 through 1-6 — every outcome above is stated in these five lessons
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