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ACT Math : Plane Geometry

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Example Questions

Example Question #1 : 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:

2.3

4.5

3.2

6.4

9.0

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:

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

$70 \%$

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

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

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

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 #22 : Squares

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

Possible Answers:

$6 \%$

$15 \%$

$9 \%$

$3 \%$

$12 \%$

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 #411 : Geometry

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

Possible Answers:

3.6

2.5

10.2

5.1

7.2

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 #171 : Quadrilaterals

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 #411 : Geometry

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 : Other Quadrilaterals

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:

10m

16m

15m

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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Example Question #412 : Act Math

Suppose a rectangle has side lengths of 7 and 3. Another rectangle has another set of side lengths that are 8 and 4. Are these similar rectangles, and why?

Possible Answers:

$\textup{There is not enough information.}$

$\textup{Yes, dimensions are only one unit more than the lengths of the other rectangle.}$

$\textup{No, they are not similar due to proportion.}$

$\textup{No, they are not similar due to area.}$

$\textup{No, they are not similar due to length.}$

Correct answer:

$\textup{No, they are not similar due to proportion.}$

Explanation:

Set up the proportion to determine if the ratios of both rectangles are equal.

If they are, then they are similar.

$\frac{7}{3} =\frac{8}{4}?$

$\frac{7}{3} \neq 2$

These are not similar rectangles since their ratios are not the same.

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Example Question #411 : Act Math

A square has a length of . What must be the length of the diagonal?

Possible Answers:

$3\sqrt{2x}$

$3\sqrt2$

$3x\sqrt2$

$9x\sqrt2$

$3x^2\sqrt2$

Correct answer:

$3x\sqrt2$

Explanation:

A square with a length of indicates that all sides are since a square has 4 equal sides. Use the Pythagorean Theorem to solve for the diagonal.

a^2+b^2=c^2

$c=\sqrt{a^2+b^2} = \sqrt{(3x)^2+(3x)^2}= \sqrt{9x^2+9x^2}= \sqrt{18x^2}= 3x\sqrt2$

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Example Question #412 : Act Math

If the rectangle has side lengths of x+3 and x+4 , what is the diagonal of the rectangle?

Possible Answers:

$\sqrt{2x^2+5}$

$x+\frac{7}{3}$

$\sqrt{2x^2+14x+25}$

$\sqrt{x^2+7x+12}$

$\sqrt{2x^2+14x+24}$

Correct answer:

$\sqrt{2x^2+14x+25}$

Explanation:

Use the Pythagorean Theorem to solve for the diagonal.

a^2+b^2=c^2

(x+3)^2+(x+4)^2=c^2

c^2 = (x^2+6x+9) + (x^2+8x+16) = 2x^2+14x+25

$c=\sqrt{2x^2+14x+25}$

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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 #22 : Squares

Example Question #411 : Geometry

Example Question #171 : Quadrilaterals

Example Question #411 : Geometry

Example Question #1 : Other Quadrilaterals

Example Question #412 : Act Math

Example Question #411 : Act Math

Example Question #412 : Act Math

All ACT Math Resources

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