Algebra: Geometry Word Problems



Geometry word problems involves geometric figures and angles described in words. You would need to be familiar with the formulas in geometry.

Making a sketch of the geometric figure is often helpful.

You can see how it is done in the following examples:
Problems involving Perimeter
Problems involving Area
Problems involving Angles

The ia also an example of a geometry word problem that uses similar triangles.

Related Topics: More Algebra Word Problems

Geometry Word Problems Involving Perimeter

Example 1:

A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side?

Solution:

Step 1: Assign variables:

Let x = length of the equal side
Sketch the figure

triangle

Step 2: Write out the formula for perimeter of triangle.

P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch.

50 = x + x + x+ 5

Combine like terms
50 = 3x + 5

Isolate variable x
3x = 50 – 5
3x = 45
x =15

Be careful! The question requires the length of the third side.

The length of third side = 15 + 5 =20

Answer: The length of third side is 20



Example 2:

Writing an equation and finding the dimensions of a rectangle knowing the perimeter and some information about the about the length and width.

The width of a rectangle is 3 feet less than its length. The perimeter of the rectangle is 110 feet. Find its dimensions.



Geometry Word Problems Involving Area

Example 1:

A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Solution:

Step 1: Assign variables:

Let x = original width of rectangle
Sketch the figure
rectangle

Step 2: Write out the formula for area of rectangle.

A = lw

Step 3: Plug in the values from the question and from the sketch.

60 = (4x + 4)(x –1)

Use distributive property to remove brackets
60 = 4x2 – 4x + 4x – 4

Put in Quadratic Form
4x2 – 4 – 60 = 0
4x2 – 64 = 0

This quadratic can be rewritten as a difference of two squares
(2x)2 – (8)2 = 0

Factorize difference of two squares
(2x)2 – (8)2 = 0
(2x – 8)(2x + 8) = 0

We get two values for x.
equations

Since x is a dimension, it would be positive. So, we take x = 4

The question requires the dimensions of the original rectangle.
The width of the original rectangle is 4.
The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the original rectangle are 4 and 16.



Example 2:

This is a geometry word problem that we can solve by writing an equation and factoring. The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. Find the height of the triangle.





Geometry Word Problems involving Angles

Example 1:

In a quadrilateral two angles are equal. The third angle is equal to the sum of the two equal angles. The fourth angle is 60° less than twice the sum of the other three angles. Find the measures of the angles in the quadrilateral.

Solution:

Step 1: Assign variables:

Let x = size of one of the two equal angles
Sketch the figure

rectangle

Step 2: Write down the sum of angles in quadrilateral.

The sum of angles in a quadrilateral is 360°

Step 3: Plug in the values from the question and from the sketch.

360 = x + x + (x + x) + 2(x + x + x + x) – 60

Combine like terms
360 = 4x + 2(4x) – 60
360 = 4x + 8x – 60
360 = 12x – 60

Isolate variable x
12x = 420
x = 35

The question requires the values of all the angles.

Substituting x for 35, you will get: 35, 35, 70, 220

Answer: The values of the angles are 35°, 35°, 70° and 220°

Example 2:

The sum of the supplement and the complement of an angle is 130 degrees. Find the measure of the angle.





Geometry Word Problems involving Similar Triangles

The following video shows an example of how to use similar triangles to solve a geometry word problem.

Raul is 6 ft tall and he notices that he casts a shadow that's 5 ft long. He then measures that the shadow cast by his school building is 30 ft long. How tall is the building?



Indirect Measurement Using Similar Triangles
This video illustrates how to find how to use the properties of similar triangles to determine the height of a tree.







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