High School Math : Quadrilaterals

Study concepts, example questions & explanations for High School Math

varsity tutors app store varsity tutors android store

Example Questions

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

What is the diagonal of a rectangle with a side length of 5 and a base of 8? (Round to the nearest tenth)

Possible Answers:

Correct answer:

Explanation:

To find the diagonal of a rectangle we must use the side lengths to create a 90 degree triangle with side lengths of 5, 8, and a hypotenuse which is equal to the diagonal.

If we have a right angle triangle and a value for two of the three side lengths, we use the Pythagorean Theorem to solve for the length of the third side.

The two side lengths that meet to form the right angle are labeled  and  which are interchangeable for each side length.

The long length connecting them is labeled  and is known as the hypotenuse.

The Pythagorean Theorem states 

Take 5 and 8 and plug them into the equation as  and  to yield 

First square the numbers 

After squaring the numbers add them together 

Once you have the sum, square root both sides 

After calculating our answer for the diagonal is .

Example Question #2 : How To Find The Length Of The Diagonal Of A Rectangle

What is the length of the diagonal of a rectangle that is 3 feet long and 4 feet wide?

Possible Answers:

7\ feet

5\ feet

8\ feet

4\ feet

6\ feet

Correct answer:

5\ feet

Explanation:

The diagonal of the rectangle is equivalent to finding the length of the hypotenuse of a right triangle with sides 3 and 4. Using the Pythagorean Theorem:

3^{2}+4^{2} = hypotenuse^{2}

25 = hypotenuse^{2}

hypotenuse = 5

Therefore the diagonal of the rectangle is 5 feet.

Example Question #3 : How To Find The Length Of The Diagonal Of A Rectangle

The length and width of a rectangle are in the ratio of 3:4. If the rectangle has an area of 108 square centimeters, what is the length of the diagonal?

Possible Answers:

12 centimeters

15 centimeters

9 centimeters

24 centimeters

18 centimeters

Correct answer:

15 centimeters

Explanation:

The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3x and 4x.

We also know the area, so we write an equation and solve for x:

(3x)(4x) = 12x= 108.

x2 = 9

x = 3

Now we can recalculate the length and the width:

length = 3x = 3(3) = 9 centimeters

width = 4x = 4(3) = 12 centimeters

Using the Pythagorean Theorem we can find the diagonal, c:

length2 + width2 = c2

92 + 12= c2 

81 + 144 = c2

225 = c2

= 15 centimeters

Example Question #6 : Quadrilaterals

A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?

 

                 

 

Possible Answers:

24

40

32

2

Correct answer:

32

Explanation:

We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)

Example Question #7 : Quadrilaterals

The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?

Possible Answers:

$4800

$5400

$2700

$1080

$1920

Correct answer:

$2700

Explanation:

The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.

Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.

10 feet = 10 x 12 inches = 120 inches

12 feet = 12 x 12 inches = 144 inches

This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.

Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.

Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.

If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.

The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.

The total cost = 1080 x 2.50 dollars = 2700 dollars.

The answer is $2700.

Example Question #8 : Quadrilaterals

Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?

Possible Answers:

10

12

4

8

6

Correct answer:

8

Explanation:

Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.

Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2l + 2w.

We also know that if s is the length of a side of a square, then the perimeter is 4s, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.

2l + 2w = 4s

If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:

Recsquare1

Recsquare2

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

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

Possible Answers:

5x + 10

6x2 + 5

5x + 5

6x2 + 10x

10(x + 1)

Correct answer:

6x2 + 10x

Explanation:

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

A = lw = (3x + 5)(2x) = 6x2 + 10x

Example Question #2 : How To Find The Area Of A Rectangle

Para-rec1

Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – x2 , and points C and D lie on the graph of y = x2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?

Possible Answers:

88

544

272

176

352

Correct answer:

176

Explanation:

Para-rec2

Para-rec3

Example Question #3 : How To Find The Area Of A Rectangle

George wants to paint the walls in his room blue.  The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor.  One gallon of paint covers 400 ft^{2} and costs $40.  One quart of paint covers 100 ft^{2} and costs $15.  How much money will he spend on the blue paint?

Possible Answers:

Correct answer:

Explanation:

The area of the walls is given by

One gallon of paint covers 400 ft^{2} and the remaining 140 ft^{2} would be covered by two quarts.

So one gallon and two quarts of paint would cost

Example Question #4 : How To Find The Area Of A Rectangle

Daisy gets new carpet for her rectangluar room.  Her floor is 21\ ft \times 24\ ft.  The carpet sells for $5 per square yard.  How much did she spend on her carpet?

Possible Answers:

\$350

\$120

\$225

\$280

\$310

Correct answer:

\$280

Explanation:

Since 3\ ft=1\ yd the room measurements are 7 yards by 8 yards.  The area of the floor is thus 56 square yards.  It would cost 5\cdot 56=\$280.

Learning Tools by Varsity Tutors