Chapter 5 Review
ISHS-Mrs. Bonn
Geometry-Semester 1
Warm-Up
Go to formulas page and write out Pythagorean
Theorem.
Leave room for the Pythagorean triples.
Example 1A: Using the Hinge Theorem and Its
Converse
Compare mÐBAC and mÐDAC
Compare the side lengths in ∆ABC and ∆ADC.
AB = AD
AC = AC
BC > DC
By the Converse of the Hinge Theorem,
mÐBAC > mÐDAC
Example 1B: Using the Hinge Theorem and Its
Converse
Compare EF and FG.
Compare the sides and angles
in ∆EFH angles in ∆GFH.
m GHF = 180° – 82° = 98°
EH = GH
FH = FH
mÐEHF > mÐGHF
By the Hinge Theorem, EF < GF.
The Pythagorean Theorem is probably the most famous
mathematical relationship. As you learned in Lesson 1-6, it
states that in a right triangle, the sum of the squares of the
lengths of the legs equals the square of the length of the
hypotenuse.
a +b =c
2
2
2
Example 1A: Using the Pythagorean Theorem
Find the value of x. Give your answer in simplest
radical form.
a2 + b2 = c2
Pythagorean Theorem
22 + 62 = x2
Substitute 2 for a, 6 for b, and x for c.
40 = x2
Simplify.
Find the positive square root.
Simplify the radical.
Example 1B: Using the Pythagorean Theorem
Find the value of x. Give your answer in simplest
radical form.
a2 + b2 = c2
(x – 2)2 + 42 = x2
x2 – 4x + 4 + 16 = x2
–4x + 20 = 0
Pythagorean Theorem
Substitute x – 2 for a, 4 for b, and x for c.
Multiply.
Combine like terms.
20 = 4x
Add 4x to both sides.
5=x
Divide both sides by 4.
Check It Out! Example 1a
Find the value of x. Give your answer in simplest
radical form.
a2 + b2 = c2
Pythagorean Theorem
42 + 82 = x2
Substitute 4 for a, 8 for b, and x for c.
80 = x2
Simplify.
Find the positive square root.
Simplify the radical.
Check It Out! Example 2
What if...? According to the recommended safety
ratio of 4:1, how high will a 30-foot ladder reach
when placed against a wall? Round to the nearest
inch.
Let x be the distance in feet from the foot of the ladder to
the base of the wall. Then 4x is the distance in feet from the
top of the ladder to the base of the wall.
A set of three nonzero whole numbers a, b, and c
such that a2 + b2 = c2 is called a Pythagorean triple.
Example 1A: Finding Side Lengths in a 45°- 45º- 90º
Triangle
Find the value of x. Give your answer in
simplest radical form.
By the Triangle Sum Theorem, the
measure of the third angle in the triangle
is 45°. So it is a 45°-45°-90° triangle with a
leg length of 8.
Example 1B: Finding Side Lengths in a 45º- 45º- 90º Triangle
Find the value of x. Give your answer in
simplest radical form.
The triangle is an isosceles right
triangle, which is a 45°-45°-90° triangle.
The length of the hypotenuse is 5.
Rationalize the denominator.
A 30°-60°-90° triangle is another special right triangle. You can
use an equilateral triangle to find a relationship between its
side lengths.
Example 3A: Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your
answers in simplest radical form.
22 = 2x
Hypotenuse = 2(shorter leg)
11 = x
Divide both sides by 2.
Substitute 11 for x.
Example 3B: Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your answers in
simplest radical form.
Rationalize the denominator.
y = 2x
Hypotenuse = 2(shorter leg).
Simplify.
Example 4: Using the 30º-60º-90º Triangle Theorem
An ornamental pin is in the shape of an
equilateral triangle. The length of each side
is 6 centimeters. Josh will attach the
fastener to the back along AB. Will the
fastener fit if it is 4 centimeters long?
Step 1 The equilateral triangle is divided into two 30°-60°-90°
triangles.
The height of the triangle is the length of the
longer leg.
Example 4 Continued
Step 2 Find the length x of the shorter leg.
6 = 2x
3=x
Hypotenuse = 2(shorter leg)
Divide both sides by 2.
Step 3 Find the length h of the longer leg.
The pin is approximately 5.2 centimeters high. So the fastener will fit.