Download Math 95 Notes Section 2.6 Formulas and Applications of Geometry

Math 95 Notes
Section 2.6
Formulas and Applications of Geometry
Objectives:
Students will solve for variables in literal equations and formulas
Students will solve geometric applications using linear equations.
Example:
P = a + b + c solve for b.
P  a  b  c To solve for b, we have to subtract a and c to the other side
Pac  aabcc
Pac  b
Example:
PV = nrt solve for t
PV  nrt To solve for t, we have to divide both sides by nr
PV nrt

nr
nr
PV
t
nr
Example:
x – 6y = -10 solve for x
x  6 y  10 To solve for x, we will add 6y to both sides
x  6 y  6 y  10  6 y
x  10  6 y or x  6 y  10
Example:
A = P(1 + rt) solve for t
A  P 1  rt  To solve for t, we will distribute first
A  P  Pr t Now we will subtract P from both sides
A  P  P  P  Pr t
A  P  Pr t Now we will divide both sides by Pr to get t by itself
A  P Pr t

Pr
Pr
A P
t
Pr
Example:
In a small rectangular wallet photo, the width is 7 cm less than the length. If the border
(perimeter) of the photo is 34 cm, find the length and width.
Width: L - 7
Length: L
In this problem, we know that the perimeter of the rectangle is 34 cm. So we are going
to use the perimeter formula for a rectangle, which is P = 2L + 2W.
P  2 L  2W
P  2 L  2( L  7)
P  2 L  2 L  14
34  4 L  14
34  14  4 L  14  14
48  4 L
48 4 L

4
4
12  L
So the Length is 12 cm and the Width is 12 – 7 or 5 cm.
Example:
The perimeter of a triangle is 16 ft. One side is 3 ft longer than the shortest side. The
third side is 1 ft longer than the shortest side. Find the lengths of all the sides.
Other side: x + 1
Longer side: x + 3
Shorter side: x
Since we know that the perimeter is 16 ft, we will use the perimeter which is
P = Side 1 + Side 2 + Side 3
P  Side 1 + Side 2 + Side 3
P  x   x  1   x  3
P  3x  4
16  3 x  4
16  4  3 x  4  4
12  3 x
12 3 x

3
3
4x
So we know that the shorter side is 4 ft, next side is 4 + 1 or 5 ft, and the longer side is
4 + 3 or 7 ft.
Example:
Two angles are complementary. One angle is 4 degrees less than three times the other
angle. Find the measure of the angles.
If two angles are complementary, that means that their angle measures add up to 90
degrees.
Angle 1: x
Angle 2: 3x – 4
x  3 x  4  90
4 x  4  90
4 x  4  4  90  4
4 x  94
4 x 94

4
4
x  23.5
So Angle 1 is 23.5 degrees and Angle 2 is 3(23.5) – 4 or 66.5 degrees.
Example:
Two angles are supplementary. One angle is 6 degrees more than four times the other.
Find the measure of the two angles.
If two angles are supplementary, that means that their angle measures add up to 180
degrees.
Angle 1: x
Angle 2: 4x + 6
x  4 x  6  180
5 x  6  180
5 x  6  180  6
5 x  174
5 x 174

5
5
x  34.8
So Angle 1 is 34.8 degrees and Angle 2 is 4(34.8) + 6 or 145.2 degrees.
Example: Find the measures of the vertical angles labeled in the figure by first solving
for y.
(3y + 26)
(5y – 54)
Since we are working with vertical angles, we know that their angle measures are equal.
3 y  26  5 y  54
3 y  3 y  26  5 y  3 y  54
26  2 y  54
26  54  2 y  54  54
80  2 y
80 2 y

2
2
40  y
So Angle 1 is 3(40) + 26 or 144 degrees and we know that the other has the same
measure since they are vertical.