Algebra: Geometry Word Problems

Geometry word problems involves geometric figures and angles descrihed 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

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

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

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

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.

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.

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

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°

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