ACT Math - Plane Geometry

The perimeter of a square is 48. What is the length of its diagonal?

$12\sqrt{2}$

$\sqrt{12}$

$24\sqrt{2}$

$48\sqrt{2}$

$2\sqrt{12}$

Explanation

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:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

Circle_graph_area3

100_π_

50_π_

25_π_

10_π_

20_π_

Explanation

Circle_graph_area2

The area of a rectangle is $\textup{in}^{2}$ and its perimeter is $\textup{in}$ . What are its dimensions?

$8\textup{ by }10\textup{ inches}$

$4\textup{ by }20\textup{ inches}$

$2\textup{ by }40\textup{ inches}$

$5.5\textup{ by } 15\textup{ inches}$

$\textup{None of the others}$

Explanation

Based on the information given to you, you know that the area could be written as:

Likewise, you know that the perimeter is:

Now, isolate one of the values. For example, based on the second equation, you know:

Dividing everything by , you get:

Now, substitute this into the first equation:

To solve for , you need to isolate all of the variables on one side:

or:

Now, factor this:

, meaning that could be either or . These are the dimensions of your rectangle.

You could also get this answer by testing each of your options to see which one works for both the perimeter and the area.

Find the total number of degrees inside a hexagon.

Explanation

To solve, simply use the following formula where is the number of sides. Thus,

$\textup{degrees}=(n-2)(180)=(6-2)(180)=4*180=720$

Find the perimeter of an equilateral triangle given side length of 2.

Explanation

To solve, simply multiply the side length by 3 since they are all equal. Thus,

Find the length of one side for a regular hexon with a perimeter of .

Explanation

Use the formula for perimeter to solve for the side length:

The long diagonal of a kite measures inches, and cuts the shorter diagonal into two pieces. If one of those pieces measures inches, what is the length in inches of the short diagonal?

Explanation

The long diagonal of a kite always bisects the short diagonal. Therefore, if one side of the bisected diagonal is inches, the entire diagonal is inches. It does not matter how long the long diagonal is.

The long diagonal of a kite measures inches, and cuts the shorter diagonal into two pieces. If one of those pieces measures inches, what is the length in inches of the short diagonal?