Download 3.3: Traditional Geometric Word Problems

Math 60
3.3: Geometric Application Problems
Section 3.3 Geometric Skills – Two Main Types
Elementary Algebra
Our Examples
Perimeter Word Problems
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Scalene Triangle
This is a triangle having all three sides of different lengths.
Isosceles Triangle
This is a triangle having 2 sides of the same length. Also, the 2
angles below these 2 sides have the same measure.
Equilateral Triangle
This is a triangle having 3 sides of the same length.
Rectangle
Length is the longer side and width is the shorter side. The
perimeter is given by: P = w + l + w + l, P = 2w + 2l, or P = 2(w + l)
Measuring lumber, fencing, etc…
Here you need to draw a good picture, label each part, and
determine which part(s) contribute toward the main equation, and
which do not.
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#5 and #6
Angle Word Problems
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Complementary Angles
Two angles that sum to 90.
Supplementary Angles
Two angles that sum to 180.
Vertical Angles
When two lines intersect forming an “X”, vertical angles occur in
pairs and are opposite each other. Vertical angles have the same
measure.
Interior Angles of a Triangle
The sum of the interior angles of a triangle is 180.
Interior Angles of a Parallelogram
A parallelogram is a four-sided figure with two pairs of opposite
sides parallel. The sum of the interior angles of a parallelogram is
360. The two smaller angles have equal measure, and the two
larger angles have equal measure.
Interior Angles of a Rhombus
A rhombus is a parallelogram with four equal sides. The sum of the
interior angles of a rhombus is 360. The two smaller angles have
equal measure, and the two larger angles have equal measure. This
is set–up and worked the same as the parallelogram problems.
Interior Angles of a Quadrilateral
A quadrilateral is a four–sided figure. The sum of the interior
angles of a quadrilateral is 360.
Interior Angles of a Trapezoid
A trapezoid is a quadrilateral with exactly one pair of opposite sides
parallel. The sum of the interior angles of a trapezoid is 360. The
measure of the two bottom angles of a trapezoid are the same and
the measure of the two top angles of a trapezoid are the same.
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#14
All triangles have interior angles that sum to 180. All quadrilaterals have interior angles that sum to 360.
3.3: Geometric Application Problems – Math 60
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Name of Figure
Picture
Scalene Triangle
Three sides of different lengths.
Isosceles Triangle
Two sides of the same length.
a
Perimeter Relationship
b
Perimeter = a + b + c
c
x
Perimeter = x + x + y
y
x
Equilateral Triangle
Three sides of the same length.
x
x
Perimeter = x + x + x
x
l
Rectangle
Opposite sides have the same length.
w
w
Perimeter
=
=
=
w+l+w+l
2w + 2l
2(w + l)
l
Name of Figure
Picture
Complementary Angles
Two angles that sum to 90.
Angular Relationship
x
x + y = 90
y
Supplementary Angles
Two angles that sum to 180.
x
Vertical Angles
Vertical angles have the same
measure.
x
y
y
x
x
Interior Angles of a Rhombus
The sum of the interior angles of a
rhombus is 360.
x + y + z = 180
z
x
Interior Angles of a Parallelogram
The sum of the interior angles of a
parallelogram is 360.
x + x + y + y = 360
y
x
y
x + x + y + y = 360
x
x
y
y
z
w
Interior Angles of a Trapezoid
The sum of the interior angles of a
trapezoid is 360.
y
x
3.3: Geometric Application Problems – Math 60
x=x
x
Interior Angles of a Triangle
The sum of the interior angles of a
triangle is 180.
Interior Angles of a Quadrilateral
The sum of the interior angles of a
quadrilateral is 360.
x + y = 180
y
y
w + x + y + z = 360
x + x + y + y = 360
x
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Directions: Set up an algebraic equation, solve, and answer the question being asked.
1.
A triangle has a perimeter of 74 inches. Find the three sides if one side is 24 inches larger than the smallest,
and the third side is three times the smallest.
2.
A slice of pie is in the shape of an isosceles triangle. If the shorter side is 8.5 inches less than twice the
longer two sides, and the perimeter of the pie is 17.5 inches, determine the length of the shorter side.
3.
An equilateral triangle has a perimeter of 37.68 cm. Find the length of each side.
4.
The length of a rectangular patio is 4 feet greater than its width. If the perimeter is 112 feet, find the patio’s
dimensions.
5.
A rectangular area is to be subdivided into three regions. Find the dimensions of this rectangle if the length
is 40 feet greater than the width, and 560 feet of fencing is available to create this area.
6.
A bookcase is to have three shelves, including the top. The height of the bookcase is to be 1 less than twice
the width. Find the width and height of this bookcase if only 19 feet of lumber is available.
7.
Two angles are complementary if the sum of their measures is 90. If angles A and B are complementary
angles, and angle A is 5 more than four times angle B, find the measure of angle A.
A
B
3.3: Geometric Application Problems – Math 60
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8.
Two angles are supplementary if the sum of their measures is 180. If angles A and B are supplementary
angles, and angle B is 15 less than twice angle A, find the measures of angles A and B.
A
9.
B
A pair of vertical angles is indicated in the figure below. Find the measure of each angle.
(4x + 2)
(5x – 25)
10.
One angle of a triangle is 10 more than the smallest, and the third angle is 5 less than three times the smallest.
Find the largest of the three angles.
11.
The smaller two angles of a parallelogram have equal measures, and the larger two angles measure five more
than six times each smaller angle. Find the measure of each angle.
12.
Each of the two larger angles of a rhombus is 6 less than twice the two smaller angles. Find the measure of
the two larger angles.
13.
The measure of one angle of a quadrilateral is 3 more than the smallest; the third angle is 5 less than eight
times the smallest; and the fourth angle is 2 more than eight times the smallest. Find the measures of all four
angles of the quadrilateral.
14.
A cosmetic sponge is in the shape of a trapezoid. The top angles measure 15 less than twice the measure of
the bottom angles. Find the measure of each angle.
3.3: Geometric Application Problems – Math 60
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Answers to Section 3.3 – Geometric Application Problems
Algebraic
Solution
Answer to the
Question Asked
x  10
10 in., 34 in., and 30 in.
x  6.5
x  12.56
w  26
4.5 inches
26 feet by 30 feet
w  80
80 feet by 120 feet
x3
3 feet by 5 feet
7.
 w   w   w   w   w  40   w  40  560
 w   w   w   2w 1   2w 1  19
 4 x  5  x  90
x  17
A = 73
8.
x   2 x  15  180
x  65
4 x  2  5x  25
 x    x  10  3x  5  180
x  27
x  35
A = 65
B = 115
110 and 110
 x    6x  5   x    6x  5  360
 x    2x  6   x    2x  6  360
 x    x  3  8x  5  8x  2  360
 x    2 x 15   x    2 x 15  360
x  25
25, 155, 25, and 155
x  62
118 and 118
x  20
20, 23, 155, and 162
x  65
65, 115, 65, and 115
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Algebraic Equation
 x    x  24   3x   74
 x    x    2x  8.5  17.5
x  x  x  37.68
 w   w  4   w   w  4  112
3.3: Geometric Application Problems – Math 60
12.56 cm
100
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