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ACT Math : Quadrilaterals

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Example Questions

Example Question #171 : Quadrilaterals

The area of a square is $64\textup{ in}^2$ , what is the perimeter of the square?

Possible Answers:

$\textup{36in}$

$\textup{32in}$

$\textup{Not enough information is provided.}$

$\textup{16in}$

24in

Correct answer:

$\textup{32in}$

Explanation:

Since the sides of a square are all the same, the area of a square can be found by $\textup{side}*\textup{side}=64\textup{in}^2.$ Therefore, the side of the square must be $8\textup{in}.$ The perimeter of a square can be found by adding up all of the four sides: $\textup{8in+8in+8in+8in=32in.}$

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Example Question #3 : How To Find The Length Of The Side Of A Square

If the area of the square is 100 square units, what is, in units, the length of one side of the square?

Possible Answers:

Correct answer:

Explanation:

$Area = Length \times Length$

100 = (Length)^2

$Length = \sqrt{100}=10$

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

In Square SQUA , $SU = \sqrt{2x}$ . Evaluate in terms of .

Possible Answers:

$\sqrt{x}$

$x\sqrt{2}$

$2\sqrt{x}$

Correct answer:

$\sqrt{x}$

Explanation:

If diagonal $\overline{SU}$ of Square SQUA is constructed, then $\bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse $SU = \sqrt{2x}$ . By the 45-45-90 Theorem, the sidelength can be calculated as follows:

$SQ = \frac{SU}{\sqrt{2}} = \frac{\sqrt{2x}}{\sqrt{2}} = \sqrt{\frac{2x}{2}} = \sqrt{x}$ .

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Example Question #4 : How To Find The Length Of The Side Of A Square

The circle that circumscribes Square SQUA has circumference 20. To the nearest tenth, evaluate .

Possible Answers:

9.0

2.3

3.2

4.5

6.4

Correct answer:

4.5

Explanation:

The diameter of a circle with circumference 20 is

$\frac{20}{\pi } \approx \frac{20 }{3.1416} \approx 6.3662$

The diameter of a circle that circumscribes a square is equal to the length of the diagonals of the square.

If diagonal $\overline{SU}$ of Square SQUA is constructed, then $\bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse approximately 6.3662. By the 45-45-90 Theorem, divide this by $\sqrt{2} \approx 1.4142$ to get the sidelength of the square:

$6.3662 \div 1.4142 \approx 4.5$

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

Rectangle RECT has area 90% of that of Square SQUA , and is 80% of . What percent of is ?

Possible Answers:

$79\frac{1}{81} \%$

$126 \frac{9}{16} \%$

$88 \frac{8} {9} \%$

$70 \%$

$112\frac{1}{2} \%$

Correct answer:

$88 \frac{8} {9} \%$

Explanation:

The area of Square SQUA is the square of sidelength , or $(SQ)^{2}$ .

The area of Rectangle RECT is $RE \cdot EC$ . Rectangle RECT has area 90% of that of Square SQUA , which is $\frac{9}{10} (SQ)^{2}$ ; is 80% of , so $RE = \frac{4}{5}SQ$ . We can set up the following equation:

$\frac{9}{10} (SQ)^{2} = RE \cdot EC$

$\frac{9}{10} (SQ)^{2} = \frac{4}{5} SQ \cdot EC$

$\frac{9}{10} \cdot SQ = \frac{4}{5} \cdot EC$

$\frac{10} {9} \cdot \frac{9}{10} \cdot SQ = \frac{10} {9} \cdot \frac{4}{5} \cdot EC$

$SQ = \frac{8} {9} \cdot EC$

As a percent, $\frac{8} {9}$ of is $\frac{8} {9} \times 100 \%= 88 \frac{8} {9} \%$

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

Reducing the area of a square by 12% has the effect of reducing its sidelength by what percent (hearest whole percent)?

Possible Answers:

$3 \%$

$6 \%$

$12 \%$

$9 \%$

$15 \%$

Correct answer:

$6 \%$

Explanation:

The area of the square was originally

$A = s^{2}$ ,

being the sidelength.

Reducing the area by 12% means that the new area is 88% of the original area, or 0.88A ; the square root of this is the new sidelength, so

$0.88A = 0.88s^{2}$

$\sqrt{0.88A} = \sqrt{0.88s^{2}} = \sqrt{0.88} \cdot \sqrt{s^{2}} \approx 0.94 s$

Each side of the new square will measure 94% of the length of the old measure - a reduction by 6%.

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

The circle inscribed inside Square SQUA has circumference 16. To the nearest tenth, evaluate .

Possible Answers:

7.2

5.1

3.6

10.2

2.5

Correct answer:

5.1

Explanation:

The diameter of a circle that is inscribed inside a square is equal to its sidelength , so all we need to do is find the diameter of the circle - which is circumference 16 divided by $\pi$ :

$\frac{16}{\pi } \approx \frac{16}{3.1416 } \approx 5.1$ .

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

Thingy

Refer to the above figure, which shows equilateral triangle $\bigtriangleup CDE$ inside Square ABCD . Also, $\overline{EF} \perp \overline{AB}$ .

Quadrilateral BCEF has area 100. Which of these choices comes closest to ?

Possible Answers:

Correct answer:

Explanation:

Let s = CD , the sidelength shared by the square and the equilateral triangle.

The area of $\bigtriangleup CDE$ is

$\frac{s^{2}\sqrt{3}}{4} = \frac{\sqrt{3}}{4} \cdot s^{2} \approx \frac{1.7321}{4} \cdot s^{2} \approx 0.4330 s^{2}$

The area of Square ABCD is $s^{2}$ .

By symmetry, $\overline{EF}$ bisects the portion of the square not in the triangle, so the area of Quadrilateral BCEF is half the difference of those of the square and the triangle. Since the area of Quadrilateral BCEF is 100, we can set up an equation:

$\frac{1}{2}\left (s ^{2} - 0.4330 s ^{2} \right ) = 100$

$\frac{1}{2}\left ( 0.5670 s ^{2} \right ) = 100$

$0.2835 s ^{2} = 100$

$s ^{2} \approx 100 \div 0.2835$

$s ^{2} \approx 352.73$

$s \approx\sqrt{ 352.73} \approx 18.7$

Of the five choices, 20 comes closest.

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Example Question #171 : Quadrilaterals

Find the length of the side of a square given its area is 144 .

Possible Answers:

Correct answer:

Explanation:

To find side length, simply take the square root of the volume. Thus,

$s=\sqrt{V}=\sqrt{144}=12$

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

The sides of rectangle A measure to , , 12m , and 12m . Rectangle B is similar to Rectangle A. The shorter sides of rectangle B measure 10m each. How long are the longer sides of Rectangle B?

Possible Answers:

15m

10m

16m

14m

Correct answer:

15m

Explanation:

In similar rectangles, the ratio of the sides must be equal.

To solve this question, the following equation must be set up:

$\small \frac{8}{12}=\frac{10}{x}$ , using $\small x$ as the variable for the missing side.

We then must cross multiply, which leaves us with:

$\small 8x=120$

Lastly, we divide both sides by 8 to solve for the missing side:

$\small \frac{8x}{8}=\frac{120}{8}$

x=15

Therefore, the longer sides of the rectangle are each 15m .

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ACT Math : Quadrilaterals

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #171 : Quadrilaterals

Example Question #3 : How To Find The Length Of The Side Of A Square

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

Example Question #4 : How To Find The Length Of The Side Of A Square

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

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

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

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

Example Question #171 : Quadrilaterals

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

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