ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #221 : Quadrilaterals

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

Possible Answers:

 

Correct answer:

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. 

 

 is equal to  inches. Since  inch is equal to  foot, the final answer is:


 inches is equal to  foot, which can be reduced to  foot

Example Question #222 : Quadrilaterals

Rhombus custom vt

Using the rhombus shown above, find the length of side .

Possible Answers:

Correct answer:

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:





Check:

Example Question #223 : Quadrilaterals

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. 

Possible Answers:

Correct answer:

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:  



where  the length of one side, and both  and  each represent one of the perpendicular diagonal lines. 

The solution is: 

Example Question #224 : Quadrilaterals

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. 

Possible Answers:

Correct answer:

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:  



where  the length of one side, and both  and  each represent one of the perpendicular diagonal lines. 

The solution is: 

Example Question #225 : Quadrilaterals

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. 

Possible Answers:

Correct answer:

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:  



where  the length of one side, and both  and  each represent one of the perpendicular diagonal lines. 

The solution is: 

Example Question #226 : Quadrilaterals

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

Possible Answers:

Correct answer:

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: 





Check:



Each side of the rhombus must equal .

Example Question #227 : Quadrilaterals

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

Possible Answers:

Correct answer:

Explanation:

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. Thus, 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.

To solve this problem, apply the perimeter formula for a rhombus: 
By applying the perimeter formula, the solution is:





Check:

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

A quadrilateral ABCD has diagonals that are perpendicular bisectors of one another. Which of the following classifications must apply to quadrilateral ABCD?

 

I. parallelogram

II. rhombus

III. square

 

 

Possible Answers:

I, II, and III

I and III only

I and II only

II and III only

Correct answer:

I and II only

Explanation:

If the diagonals of a quadrilateral are perpendicular bisectors of one another, then the quadrilateral must be a rhombus, but not necessarily a square. Since all rhombi are also parallelograms, quadrilateral ABCD must be both a rhombus and parallelogram.

 

 

Example Question #32 : Rhombuses

Rhomboids

Are the two rhombuses in the above picture similar?

Possible Answers:

Maybe

Not enough information to decide.

No

Yes

Correct answer:

No

Explanation:

Two shapes are similar to each other if they are the same except for differences in scaling. This means that all of the angles of one shape have to be the same as the angles of the other shape, and all the sides have to be proportional to each other. For example, if we have two rectangles A and B, where A has side lengths 2 and 4, and B has side lengths 5 and 10, A and B are similar because the ratio between 2 and 4 is the same as the ratio between 5 and 10, and also the ratio between 2 and 5 is the same as the ratio between 4 and 10. Everything is proportional (and all the angles are the same) so the two rectangles A and B are similar.

The two rhombuses in our figure are not similar to each other, because we can see the larger angle in the rhombus on the left is not the same as the larger angle in the rhombus on the right. If we scaled the larger rhombus down and tried to cover up the smaller rhombus with it, it wouldn't work because they aren't proportional to each other.

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

In the figure below,  and  are radii of the circle. If the radius of the circle is 8 units, how long, in units, is chord ?

Circle

Possible Answers:

Correct answer:

Explanation:

The radii and the chord form right triangle . Use the Pythagorean Theorem to find the length of .

 

Learning Tools by Varsity Tutors