1. 2
Equations, Inequalities, and Applications

2. 2.5 Formulas and Additional Applications from Geometry
Objectives
1. Solve a formula for one variable, given the values of the other variables. 2. Use a formula to solve an applied problem. 3. Solve problems involving vertical angles and straight angles. 4. Solve a formula for a specified variable.

3. Solve a Formula for One Variable
Example 1
Find the value of the remaining variable.
P = 2L + 2W; P = 52; L = 8
P = 2L + 2W
Check: 2 · · 18 = 52
52 = 2 · 8 + 2W
= 52
52 = 52
52 = W
–16
–16
36 = 2W
18 = W

4. Solve a Formula for One Variable
AREA FORMULAS
Triangle
A = ½bh
Rectangle
A = LW
Trapezoid
A = ½h(b + B)
b = base
h = height h b L
L = Length
W = Width W
h = height
b = small base
B = large base b h B

5. Use a Formula to Solve an Applied Problem
Example 2
The area of a rectangular garden is 187 in2 with a width of 17 in. What is the length of the garden?
A = LW
187 = L · 17
11 = L 17
The length is 11 in.
Check: 17 · 11 = 187

6. Use a Formula to Solve an Applied Problem
Example 3
Bob is working on a sketch for a new underwater vehicle (UV), shown below. In his sketch, the bottom of the UV is 10 ft long, the top is 8 ft long, and the area is 63 ft2. What is the height of his UV?
A = ½h(b + B)
The height of the UV is 7ft. 8
63 = ½h(8 + 10) h 10
63 = ½h(18)
Check:
½ · 7 · 18 = 63
63 = 9h
7 = h

7. Solve Problems Involving Vertical and Straight Angles2 1 3 4
The figure shows two intersecting lines forming angles that are numbered: , , , and . 1 2 3 4
Angles 1 and 3 lie “opposite” each other. They are called vertical angles. Another pair of vertical angles is 2 and 4.. Vertical angles have equal measures.
Now look at angles 1 and 2. When their measures are added, we get the measure of a straight angle, which is 180°. There are three other such pairs of angles: 2 and 3, 3 and 4 and 4 and 1.

8. Solve Problems Involving Vertical and Straight Angles
Example 5a
Find the measure of each marked angle below.
Since the marked angles are vertical angles, they have equal measures.
3x – 4 = 5x – 40
Thus, both angles are
(3x – 4)°
(5x – 40)°
–3x
–3x
–4 = 2x – 40
3 · 18 – 4 = 50°
+ 40
+ 40
Check:
5 · 18 – 40 = 50°
CAUTION
Here, the answer was
not the value of x!
36 = 2x
18 = x

9. Solve Problems Involving Vertical and Straight Angles
Example 5b
Find the measure of each marked angle below.
Since the marked angles are straight angles, their sum will be 180°.
(6x – 10)°
(x + 15)°
6x – 10 + x + 15 = 180
Thus, the angles are
7x + 5 = 180
6 · 25 – 10 = 140°
and
= 40°
– 5
– 5
Check:
140° + 40° = 180°
7x = 175
x = 25

10. Solve a Formula for a Specified Variable
Example 6
Solve A = ½bh for b.
The goal is to get b alone on one side of the equation.
2 ·
A = ½bh
· 2
2A = bh