All ACT Math Resources
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 .
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 ?
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)?
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 .
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
Refer to the above figure, which shows equilateral triangle inside Square . Also, .
Quadrilateral has area 100. Which of these choices comes closest to ?
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 .
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?
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?
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?
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?
Use the Pythagorean Theorem to solve for the diagonal.