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The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

26 ft

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The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

29 m

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The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. the length and width of the pool
Step 3. Name. Choose a variable to represent it.
The length is 15 feet more than the width.
Let W = width
W + 15 = length
Step 4. Translate.
Write the appropriate formula and substitute.
.
Step 5. Solve the equation. .
Step 6. Check:
.
Step 7. Answer the question. The length of the pool is 45 feet and the width is 30 feet.
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The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

30 ft, 70 ft

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The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

60 yd, 90 yd

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Use the properties of triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in [link] , we’ve labeled the length b and the width h , so it’s area is b h .

A rectangle is shown. The side is labeled h and the bottom is labeled b. The center says A equals bh.
The area of a rectangle is the base, b , times the height, h .

We can divide this rectangle into two congruent triangles ( [link] ). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or 1 2 b h . This example helps us see why the formula for the area of a triangle is A = 1 2 b h .

A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says “Area of each triangle,” and shows the equation A equals one-half bh.
A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

The formula for the area of a triangle is A = 1 2 b h , where b is the base and h is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a 90° angle with the base. [link] shows three triangles with the base and height of each marked.

Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.
The height h of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a 90° angle with the base.

Triangle properties

For any triangle Δ A B C , the sum of the measures of the angles is 180° .

m A + m B + m C = 180°

The perimeter of a triangle is the sum of the lengths of the sides.

P = a + b + c

The area of a triangle is one-half the base, b , times the height, h .

A = 1 2 b h
A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.

Find the area of a triangle whose base is 11 inches and whose height is 8 inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. the area of the triangle
Step 3. Name. Choose a variable to represent it. let A = area of the triangle
Step 4. Translate.
Write the appropriate formula.
Substitute.

.
Step 5. Solve the equation. .
Step 6. Check:
.
Step 7. Answer the question. The area is 44 square inches.
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Practice Key Terms 6

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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