In kite ABCD, diagonals AC and BD intersect perpendicularly at point P. If AP = 9 centimet… — ACT

In kite $ABCD$ , diagonals $AC$ and $BD$ intersect perpendicularly at point $P$ . If $AP = 9$ centimeters, $PC = 16$ centimeters, and $BP = PD = 12$ centimeters, what is the perimeter, in centimeters, of kite $ABCD$ ?

Answer: 70 centimeters

Answer

The perimeter of kite

ABCD

is 70 centimeters.

The diagonals of a kite intersect at right angles (

90^\circ

). Applying the Pythagorean theorem to right triangle

APB

with legs

9

cm and

12

cm gives hypotenuse

AB = 15

cm. Applying the Pythagorean theorem to right triangle

BPC

with legs

16

cm and

12

cm gives hypotenuse

BC = 20

cm. Since a kite has two pairs of equal adjacent sides (

AB = AD = 15

cm and

BC = CD = 20

cm), the total perimeter is

15 + 15 + 20 + 20 = 70

cm.

Step-by-Step Solution

Identify right triangles formed by the perpendicular diagonals

Four right triangles are formed:

\triangle APB

\triangle BPC

\triangle CPD

, and

\triangle DPA

Diagonals of a kite are perpendicular to each other.

Calculate upper side length

AB

AB = \sqrt{9^2 + 12^2} = \sqrt{225} = 15

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to right triangle

APB

Calculate lower side length

BC

BC = \sqrt{16^2 + 12^2} = \sqrt{400} = 20

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to right triangle

BPC

Compute the total perimeter

Perimeter

= 2(15) + 2(20) = 30 + 40 = 70

A kite has two pairs of congruent adjacent sides (

AD = AB

and

CD = BC

Key Concept

Perpendicular diagonals and side length properties of a kite

Alternative Method

Instead of calculating all four sides individually, calculate one side from each distinct right triangle (

15

cm and

20

cm) and multiply their sum by

2

, using the property that a kite has two symmetric pairs of congruent adjacent sides:

2 \times (15 + 20) = 70

cm.

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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