High School Math : Rectangles

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #1 : How To Find If Rectangles Are Similar

Two rectangles are similar. The perimeter of the first rectangle is 36. The perimeter of the second is 12. If the base length of the second rectangle is 4, what is the height of the first rectangle?

Possible Answers:

6

2

10

4

8

Correct answer:

6

Explanation:

Solve for the height of the second rectangle.

Perimeter = 2B + 2H

12 = 2(4) + 2H

12 = 8 + 2H

4 = 2H

H = 2

If they are similar, then the base and height are proportionally equal.

B1/H1 = B2/H2

4/2 = B2/H2

2 = B2/H2

B2 = 2H2

Use perimeter equation then solve for H:

Perimeter = 2B + 2H

36 = 2 B2 + 2 H2

36 = 2 (2H2) + 2 H2

36 = 4H2 + 2 H2

36 = 6H2

H2 = 6

Example Question #1 : How To Find The Perimeter Of A Rectangle

A rectangle has a width of 2x. If the length is five more than 150% of the width, what is the perimeter of the rectangle?

Possible Answers:

5x + 10

6x2 + 10x

10(x + 1)

6x2 + 5

5x + 5

Correct answer:

10(x + 1)

Explanation:

Given that w = 2x and l = 1.5w + 5, a substitution will show that l = 1.5(2x) + 5 = 3x + 5.  

P = 2w + 2l = 2(2x) + 2(3x + 5) = 4x + 6x + 10 = 10x + 10 = 10(x + 1)

Example Question #1 : Rectangles

What is the perimeter of the below rectangle in simplest radical form?

 

                                     Act_math_159_15

Possible Answers:

5√3

4√3 + 2√27

7√27

10√3

Correct answer:

10√3

Explanation:

The perimeter of a figure is the sum of the lengths of all of its sides. The perimeter of this figure is √27 + 2√3 + √27 + 2√3. But, √27 = √9√3 = 3√3 . Now all of the sides have the same number underneath of the radical symbol (i.e. the same radicand) and so the coefficients of each radical can be added together. The result is that the perimeter is equal to 10√3.

 

 

Example Question #1 : How To Find The Perimeter Of A Rectangle

A rectangle has an area of 56 square feet, and a width of 4 feet. What is the perimeter, in feet, of the rectangle?

Possible Answers:
30
14
120
36
28
Correct answer: 36
Explanation:

Divide the area of the rectangle by the width in order to find the length of 14 feet. The perimeter is the sum of the side lengths, which in this case is 14 feet + 4 feet +14 feet + 4 feet, or 36 feet.

Example Question #1 : How To Find The Perimeter Of A Rectangle

The length of a rectangle is 3 more inches than its width.  The area of the rectangle is 40 in2.  What is the perimeter of the rectangle?

Possible Answers:

26 in.

40 in.

None of the answers are correct

18 in.

34 in.

Correct answer:

26 in.

Explanation:

Area of rectangle:  A = lw

Perimeter of rectangle:  P = 2l + 2w

w = width and l = w + 3

So A = w(w + 3) = 40 therefore w2 + 3w – 40 = 0

Factor the quadratic equation and set each factor to 0 and solve.

w2 + 3w – 40 = (w – 5)(w + 8) = 0 so w = 5 or w = -8. 

The only answer that makes sense is 5.  You cannot have a negative value for a length.

Therefore, w = 5 and l = 8, so P = 2l + 2w = 2(8) + 2(5) = 26 in.

Example Question #1 : How To Find The Perimeter Of A Rectangle

Rectangle

The length of a rectangle is  and the width is . What is its perimeter?

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

Perimeter of a quadrilateral is found by adding up the lengths of its sides. The formula for the perimeter of a rectangle is .

Example Question #131 : Quadrilaterals

Kayla took 25 minutes to walk around a rectangular city block. If the block's width is 1/4 the size of the length, how long would it take to walk along one length?

Possible Answers:

2.5 minutes

8 minutes

10 minutes

7 minutes

Correct answer:

10 minutes

Explanation:

Leaving the width to be x, the length is 4x. The total perimeter is 4x + 4x + x + = 10x.

We divide 25 by 10 to get 2.5, the time required to walk the width.  Therefore the time required to walk the length is (4)(2.5) = 10.

Example Question #1 : How To Find The Perimeter Of A Rectangle

Robert is designing a rectangular garden. He wants the area of the garden to be 9 square meters. If the length of the lot is going to be three meters less than twice the width, what will the perimeter of the lot be in meters?

 

Possible Answers:

12

10

3

6

1.5

Correct answer:

12

Explanation:

Let l be the length of the garden and w be the width. 

By the specifications of the problem, l = 2w-3.

Plug this in for length in the area formula:

A = l  x w = (2w - 3) x w = 9

Solve for the width:

2w²- 3w - 9 =0 

(2w + 3)(w - 3) = 0

w is either 3 or -3/2, but we can't have a negative width, so w = 3.

If w = 3, then length = 2(3) - 3 = 3.

Now plug the width and length into the formula for perimeter:

P = 2 l + 2w = 2(3) + 2(3) = 12

 

 

Example Question #1 : Rectangles

A rectangular garden has an area of . Its length is  meters longer than its width. How much fencing is needed to enclose the garden?

Possible Answers:

Correct answer:

Explanation:

We define the variables as  and .

We substitute these values into the equation for the area of a rectangle and get

 or 

Lengths cannot be negative, so the only correct answer is . If , then

Therefore, .

Example Question #1 : How To Find The Length Of The Side Of A Rectangle

 

 

 

The two rectangles shown below are similar. What is the length of EF?

 Sat_mah_166_02

Possible Answers:

10

8

6

5

Correct answer:

10

Explanation:

When two polygons are similar, the lengths of their corresponding sides are proportional to each other.  In this diagram, AC and EG are corresponding sides and AB and EF are corresponding sides. 

To solve this question, you can therefore write a proportion:

AC/EG = AB/EF ≥ 3/6 = 5/EF

From this proportion, we know that side EF is equal to 10.

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