ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

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

Possible Answers:

Correct answer:

Explanation:

The diameter of a circle with circumference 20 is

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

If diagonal  of Square  is constructed, then  is a 45-45-90 triangle with hypotenuse approximately 6.3662. By the 45-45-90 Theorem, divide this by  to get the sidelength of the square:

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

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

Possible Answers:

Correct answer:

Explanation:

The area of Square  is the square of sidelength , or .

The area of Rectangle  is . Rectangle  has area 90% of that of Square , which is ;   is 80% of , so . We can set up the following equation: 

As a percent,  of  is 

 

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:

Correct answer:

Explanation:

The area of the square was originally 

 being the sidelength.

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

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

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

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

Possible Answers:

Correct answer:

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 :

.

 

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

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Refer to the above figure, which shows equilateral triangle  inside Square . Also, .

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

Possible Answers:

Correct answer:

Explanation:

Let , the sidelength shared by the square and the equilateral triangle.

The area of  is

The area of Square  is .

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

Of the five choices, 20 comes closest.

Example Question #411 : Plane Geometry

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

Possible Answers:

Correct answer:

Explanation:

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

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

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

Possible Answers:

Correct answer:

Explanation:

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

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

, using  as the variable for the missing side. 

We then must cross multiply, which leaves us with:

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

Therefore, the longer sides of the rectangle are each .

Example Question #181 : Quadrilaterals

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:

Correct answer:

Explanation:

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

If they are, then they are similar.  

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

Example Question #181 : Quadrilaterals

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

Possible Answers:

Correct answer:

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.

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

If the rectangle has side lengths of  and , what is the diagonal of the rectangle?

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem to solve for the diagonal.

 

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