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More Algebra Word Problems

Geometry Games

Making a sketch of the geometric figure is often helpful.

You can see how to solve geometry word problems in the following examples:

Problems involving Perimeter

Problems involving Area

Problems involving Angles

There is also an example of a geometry word problem that uses similar triangles.

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 = 3*x* + 5

Isolate variable *x
*3

3

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.

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 = (4*x* + 4)(*x* –1)

Use distributive
property to remove brackets

60 = 4*x*^{2} – 4*x* + 4*x* – 4

Put in Quadratic Form

4*x*^{2} – 4 – 60 = 0

4*x*^{2} – 64 = 0

This quadratic can be rewritten as a difference of two squares

(2*x*)^{2} – (8)^{2} = 0

Factorize difference
of two squares

(2*x*)^{2} – (8)^{2} = 0

(2*x* – 8)(2*x* + 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.

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.

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 = 4*x* + 2(4*x*) – 60

360 = 4*x* + 8*x* – 60

360 = 12*x* – 60

Isolate variable *x
*12

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.

1. Given that triangle ABC is similar to triangle DEF, solve for x and y.

2. The extendable ramp shown below is used to move crates of fruit to loading docks of different heights.

When the horizontal distance AB is 4 feet, the height of the loading dock, BC, is 2 feet. What is the height of the loading dock, DE?

3. Triangles ABC and A'B'C' are similar figures. Find the length AB.

Example:

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?

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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