SAT Mc Graw Hill 2011

CHAPTER 10 / ESSENTIAL GEOMETRY SKILLS 371

The Pythagorean Theorem

The Pythagorean theoremsays that in any right

triangle, the sum of the squares of the two

shorter sides is equal to the square of the longest

side. If you know two sides of any right trian-

gle, the Pythagorean theorem can always be

used to find the third side.

Example:

In the figure below, what is x?

Pythagorean theorem: 92 +x^2 = 152

Simplify: 81 +x^2 = 225

Subtract 81: x^2 = 144

Take the square root: x= 12

You can also use the modified Pythagorean

theorem to find whether a triangle is acute or

obtuse.

If (side 1 )^2 +(side 2 )^2 < (longest side)^2 , the triangle is

obtuse.(If the stick gets bigger, the alligator’s

mouth gets wider!)

If (side 1 )^2 +(side 2 )^2 > (longest side)^2 , the triangle is

acute.(If the stick gets smaller, the alligator’s

mouth gets smaller!)

Special Right Triangles

Certain special right triangles show up frequently on
the SAT. If you see that a triangle fits one of these pat-
terns, it may save you the trouble of using the
Pythagorean Theorem. But be careful: you must
know two of three “parts” of the triangle in order to
assume the third part.

x

15

9

c^2

c

b^2

a^2

a

b

a^2 + b^2 = c^2

3-4-5 triangles More accurately, these can

be called 3x-4x-5xtriangles because the multi-

ples of 3-4-5 also make right triangles. Notice

that the sides satisfy the Pythagorean theorem.

5-12-13 triangles Likewise, these can be

called 5x-12x-13xtriangles because the multi-

plesof 5-12-13 also make right triangles. No-

tice that the sides satisfy the Pythagorean

theorem.

45 °-45°-90°triangles These triangles can be

thought of as squares cut on the diagonal.This

shows why the angles and sides are related the

way they are. Notice that the sides satisfy the

Pythagorean theorem.

x x^2

45 °

45 °

x

5

12

2.5 13

6

6.5

3

4

5

16

20 12

Lesson 3: The Pythagorean Theorem