# ACT Math : Kites

## Example Questions

### Example Question #11 : How To Find An Angle In A Kite

A kite has one set of opposite interior angles where the two angles measure  and , respectively. Find the measurement for one of the two remaining interior angles in this kite.

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

degrees, where  is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:  degrees  degrees  degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between  degrees and the non-congruent opposite angles sum by :

This means that  is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is:

### Example Question #11 : How To Find An Angle In A Kite

A kite has one set of opposite interior angles where the two angles measure  and , respectively. Find the measurement of the sum of the two remaining interior angles.

Explanation:

The sum of the interior angles of any polygon can be found by applying the formula:

degrees, where  is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:  degrees  degrees  degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles

To find the sum of the remaining two angles, determine the difference between  degrees and the sum of the non-congruent opposite angles.

The solution is:

This means that  is the sum of the remaining two opposite angles.

### Example Question #11 : How To Find An Angle In A Kite

If the diagonals of the quadrilateral above were drawn in the figure, they would form four 90 degree angles at the center. In degrees, what is the value of ?

Explanation:

A quadrilateral is considered a kite when one of the following is true:

(1) it has two disjoint pairs of sides are equal in length or

(2) one diagonal is the perpendicular bisector of the other diagonal. Given the information in the question, we know (2) is definitely true.

To find  we must first find the values of all of the angles.

The sum of angles within any quadrilateral is 360 degrees.

Therefore .

To find :

### Example Question #14 : How To Find An Angle In A Kite

In a particular kite, one angle that lies between congruent sides measures , and one angle that lies between non-congruent sides measures . What is the measure of the angle opposite the  angle?

Explanation:

One of the rules governing kites is that the angles which lie between non-congruent sides are congruent to each other. Thus, we know one of the missing angles is also . Since all angles in a quadrilateral must sum to , we know that the other missing angle is

### Example Question #1 : Kites

Using the kite shown above, find the length of the red (vertical) diagonal.

Explanation:

In order to solve this problem, first observe that the red diagonal line divides the kite into two triangles that each have side lengths of  and  Notice, the hypotenuse of the interior triangle is the red diagonal. Therefore, use the Pythagorean theorem: , where  the length of the red diagonal.

The solution is:

### Example Question #3 : Plane Geometry

A kite has two perpendicular interior diagonals. One diagonal is twice the length of the other diagonal. The total area of the kite is . Find the length of each interior diagonal.

Explanation:

To solve this problem, apply the formula for finding the area of a kite:

However, in this problem the question only provides information regarding the exact area. The lengths of the diagonals are represented as a ratio, where

Therefore, it is necessary to plug the provided information into the area formula. Diagonal  is represented by  and diagonal .

The solution is:

Thus, if , then diagonal  must equal

### Example Question #4 : Plane Geometry

A kite has two perpendicular interior diagonals. One diagonal has a measurement of  and the area of the kite is . Find the length of the other interior diagonal.

Explanation:

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

### Example Question #251 : Geometry

A kite has two perpendicular interior diagonals. One diagonal has a measurement of  and the area of the kite is . Find the length of the other interior diagonal.

Explanation:

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

### Example Question #6 : Plane Geometry

A kite has two perpendicular interior diagonals. One diagonal has a measurement of  and the area of the kite is . Find the sum of the two perpendicular interior diagonals.

Explanation:

First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Therefore, the sum of the two diagonals is:

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

A kite has two perpendicular interior diagonals. One diagonal has a measurement of  and the area of the kite is . Find the sum of the two perpendicular interior diagonals.

Explanation:

You must find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Therefore, the sum of the two diagonals is: