Week 27: The Pythagorean theorem, distance, and slope

Field Book

97 to 100

Due

Test 24 due

Method

Paper first, every step

The most famous equation in school mathematics shows up this week, and then immediately does a second job. The distance between two points on a graph is the Pythagorean theorem wearing a coat: the horizontal gap and the vertical gap are the two legs, and the distance you want is the hypotenuse. Same theorem, different clothes.

What you have to be able to do

Use the Pythagorean theorem and recognize the common triples
Find the distance between two points on the coordinate plane
Find the slope of a line through two points

The procedures

The Pythagorean theorem

a squared plus b squared equals c squared, and c is always the side across from the right angle.
Put the two legs in for a and b, solve for c squared, then take the square root.
Worth memorizing: 3-4-5, 5-12-13, and 8-15-17, along with any multiple of those.

Distance between two points

Find the horizontal gap by subtracting the x values, and the vertical gap by subtracting the y values.
Those two gaps are the legs of a right triangle.
Apply the Pythagorean theorem. The distance is the hypotenuse.

Slope from two points

Slope is the change in y divided by the change in x, rise over run.
Subtract the points in the same order on the top and the bottom. Reversing one and not the other flips the sign.
A positive slope climbs left to right, a negative one falls.

Worked

WORKED

A right triangle has legs 6 and 8. Find the hypotenuse.
36 + 64 = 100, so c squared is 100.
ANSWER: c = 10, and this is the 3-4-5 triple doubled

WORKED

Find the distance and the slope between (1, 2) and (4, 6).
Horizontal gap 3, vertical gap 4.
Distance: 9 + 16 = 25, so the distance is 5. Slope: rise over run is 4 over 3.
ANSWER: distance 5, slope 43

Terms

Term	Meaning
hypotenuse	the longest side of a right triangle, across from the right angle
Pythagorean triple	three whole numbers that satisfy the theorem exactly, like 3, 4, 5
slope formula	the change in y over the change in x between two points

Check yourself

1. Legs 5 and 12. Find the hypotenuse.

2. Distance between (0, 0) and (3, 4)

3. Slope through (2, 1) and (5, 7)

reveal answers

13 · 5 · 2

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.