ACT Math - Rhombuses | Practice Hub

1

A rhombus has a perimeter of inches. Find the length for one side of the rhombus.

$\frac{2}{3}\textup{ foot}$

$4\textup{ inches}$

$\frac{1}{4}\textup{ foot}$

$\frac{3}{4}\textup{ foot}$

$9\textup{ inches}$

Explanation

To solve this problem, apply the perimeter formula for a rhombus: .

Note that the perimeter formula for a rhombus is the same formula used to find the perimeter of a square. This is because both shapes, by definition, have equivalent sides. The total perimeter is the sum of all sides.

The primary differentiation between rhombuses and squares is that latter must have four interior right angles. Although the four interior angles of a rhombus must also equal a sum of 360 degrees, the interior angles inside of a rhombus do not need to be right angles. Instead, the adjacent interior angles of a rhombus must be supplementary angles.

By applying the perimeter formula, the solution is:

, where is equal to one of the four sides.

$s=\frac{32}{4}=8$

is equal to inches. Since inch is equal to $\frac{1}{12}$ foot, the final answer is:

inches is equal to $\frac{8}{12}$ foot, which can be reduced to $\frac{8\div 4}{12\div 4}=\frac{2}{3}$ foot

2

A rhombus has a perimeter of inches. Find the length for one side of the rhombus.

$\frac{2}{3}\textup{ foot}$

$4\textup{ inches}$

$\frac{1}{4}\textup{ foot}$

$\frac{3}{4}\textup{ foot}$

$9\textup{ inches}$

Explanation

To solve this problem, apply the perimeter formula for a rhombus: .

Note that the perimeter formula for a rhombus is the same formula used to find the perimeter of a square. This is because both shapes, by definition, have equivalent sides. The total perimeter is the sum of all sides.

The primary differentiation between rhombuses and squares is that latter must have four interior right angles. Although the four interior angles of a rhombus must also equal a sum of 360 degrees, the interior angles inside of a rhombus do not need to be right angles. Instead, the adjacent interior angles of a rhombus must be supplementary angles.

By applying the perimeter formula, the solution is:

, where is equal to one of the four sides.

$s=\frac{32}{4}=8$

is equal to inches. Since inch is equal to $\frac{1}{12}$ foot, the final answer is:

inches is equal to $\frac{8}{12}$ foot, which can be reduced to $\frac{8\div 4}{12\div 4}=\frac{2}{3}$ foot

3

A rhombus has a perimeter of . Find the length for one side of the rhombus.

Explanation

To solve this problem, apply the perimeter formula for a rhombus: .
Note that the perimeter formula for a rhombus is the same formula used to find the perimeter of a square. This is because both shapes, by definition, have equivalent sides. The total perimeter is the sum of all sides.

The primary differentiation between rhombuses and squares is that latter must have four interior right angles. Although the four interior angles of a rhombus must also equal a sum of 360 degrees, the interior angles inside of a rhombus do not need to be right angles. Instead, the adjacent interior angles of a rhombus must be supplementary angles.

By applying the perimeter formula, the solution is:

$s=\frac{68}{4}=17$

Check:

$17\times4=68$

Each side of the rhombus must equal .

4

A rhombus has a perimeter of . Find the length for one side of the rhombus.

Explanation

To solve this problem, apply the perimeter formula for a rhombus: .
Note that the perimeter formula for a rhombus is the same formula used to find the perimeter of a square. This is because both shapes, by definition, have equivalent sides. The total perimeter is the sum of all sides.

The primary differentiation between rhombuses and squares is that latter must have four interior right angles. Although the four interior angles of a rhombus must also equal a sum of 360 degrees, the interior angles inside of a rhombus do not need to be right angles. Instead, the adjacent interior angles of a rhombus must be supplementary angles.

By applying the perimeter formula, the solution is:

$s=\frac{68}{4}=17$

Check:

$17\times4=68$

Each side of the rhombus must equal .

5

Find the area of a rhombus if the diagonals lengths are and .

Explanation

Write the formula for the area of a rhombus:

$A=\frac{d_1 \cdot d_2}{2}$

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20:cm \cdot 40:cm}{2} =\frac{800:cm^2}{2} = 400:cm^2$

6

If a side length of a rhombus is half, what is the perimeter of the rhombus?

$\frac{\sqrt2}{2}$

$\frac{1}{8}$

Explanation

Write the formula to find the perimeter of a rhombus.

Substitute the side length and simplify.

$P=4(\frac{1}{2})=2$

7

A rhombus has an interior angle with a measurement of $73$^\circ$$ . Find the measurement for one of the angles that are adjacent to the angle that has a measurement of $73$^\circ$$ .

$107$^\circ$$

$117$^\circ$$

$34$^\circ$$

$68$^\circ$$

$180$^\circ$$

Explanation

A rhombus must have two sets of equivalent opposite interior angles, and a rhombus must have two sets of adjacent interior angles. The adjacent interior angles must be supplementary—meaning they have a sum total of $180$^\circ$$ .

If a rhombus has an interior angle that has a measurement of $73$^\circ$$ , the adjacent angle must equal:

$180-73=107$^\circ$$

8

If a side length of a rhombus is half, what is the perimeter of the rhombus?

$\frac{\sqrt2}{2}$

$\frac{1}{8}$

Explanation

Write the formula to find the perimeter of a rhombus.

Substitute the side length and simplify.

$P=4(\frac{1}{2})=2$

9

Find the area of a rhombus if the diagonals lengths are and .

Explanation

Write the formula for the area of a rhombus:

$A=\frac{d_1 \cdot d_2}{2}$

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20:cm \cdot 40:cm}{2} =\frac{800:cm^2}{2} = 400:cm^2$

10

A rhombus has two perpendicular interior diagonal lines, each with endpoints that are the vertex of opposite interior angles. The diagonals have lengths of and . Find the length for one side of the rhombus.

Explanation

This problem provides the lengths of the two perpendicular interior diagonal lines in the rhombus. To use this information to find the length of one side of the rhombus, apply the formula:

$s=\frac{\sqrt{a^2+b^2}}{2}$