ACT Math : Plane Geometry
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Example Questions
Example Question #1 : How To Find If Rectangles Are Similar
Two rectangles are similar. The perimeter of the first rectangle is 36. The perimeter of the second is 12. If the base length of the second rectangle is 4, what is the height of the first rectangle?
4
6
10
2
8
6
Solve for the height of the second rectangle.
Perimeter = 2B + 2H
12 = 2(4) + 2H
12 = 8 + 2H
4 = 2H
H = 2
If they are similar, then the base and height are proportionally equal.
B1/H1 = B2/H2
4/2 = B2/H2
2 = B2/H2
B2 = 2H2
Use perimeter equation then solve for H:
Perimeter = 2B + 2H
36 = 2 B2 + 2 H2
36 = 2 (2H2) + 2 H2
36 = 4H2 + 2 H2
36 = 6H2
H2 = 6
Example Question #1 : How To Find If Rectangles Are Similar
Two rectangles are similar. The first rectangle has a perimeter of 
1. Create a proportion comparing the two given rectangles:
2. Solve for the length of the second rectangle by cross-multiplying:
Example Question #2 : How To Find If Rectangles Are Similar
The width of a rectangle is 5 times the length of the rectangle. The width of the rectangle is 30. What is the perimeter of the rectangle?
If the width is 5 times the length, and the width is 30, then the length is 6. The perimeter of a rectangle is 2 x width + 2 x length. 2 x 30 + 2 x 6 = 72.
Example Question #1 : How To Find The Length Of The Diagonal Of A Square
The perimeter of a square is 48. What is the length of its diagonal?
Perimeter = side * 4
48 = side * 4
Side = 12
We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.
Therefore, we can use the Pythagorean Theorem to solve for the diagonal:
Example Question #143 : Quadrilaterals
What is the area of a square with a diagonal of length 
Based on the data provided, the square could be drawn like this:
Based on this, you can use the Pythagorean theorem to find 
From this, you know that
Since the area is equal to 
Example Question #3 : How To Find The Length Of The Diagonal Of A Square
The perimeter of a square is 
From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,
Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.
Example Question #381 : Plane Geometry
Find the diagonal of a square with side length 
To solve, simply realize the triangle that is made by 
Example Question #2 : Squares
Find the length of a diagonal of a square whose side length is 
To solve, simply remember that the diagonal forms an isoceles right triangle. Thus, the diagonal is:
Example Question #4 : How To Find The Area Of A Square
How much more area does a square with a side of 2r have than a circle with a radius r? Approximate π by using 22/7.
1/7 square units
4/7 square units
6/7 square units
12/14 square units
6/7 square units
The area of a circle is given by A = πr2 or 22/7r2
The area of a square is given by A = s2 or (2r)2 = 4r2
Then subtract the area of the circle from the area of the square and get 6/7 square units.
Example Question #33 : Squares
If the perimeter of a square is 44 centimeters, what is the area of the square in square centimeters?
Since the square's perimeter is 44, then each side is 
Then in order to find the area, use the definition that the
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