Kite WXYZ has perpendicular diagonals WY and XZ that intersect at point M. Diagonal WY bis… — ACT

Kite $WXYZ$ has perpendicular diagonals $WY$ and $XZ$ that intersect at point $M$ . Diagonal $WY$ bisects diagonal $XZ$ such that $XM = MZ = 8$ centimeters. If $WM = 6$ centimeters and $MY = 15$ centimeters, what is the perimeter of kite $WXYZ$ , in centimeters?

A
27
B
46
54Answer
D
68
E
74

Answer

The perimeter of kite

WXYZ

is 54 centimeters.

The correct answer is 54. The perpendicular diagonals of a kite form four interior right triangles. Using the legs

WM = 6

cm and

XM = 8

cm, the upper side

WX

\sqrt{6^2 + 8^2} = 10

cm. Using the legs

MY = 15

cm and

XM = 8

cm, the lower side

YX

\sqrt{15^2 + 8^2} = 17

cm. Summing all four outer sides (

10 + 10 + 17 + 17

) gives a total perimeter of 54 cm.

Step-by-Step Solution

Identify key geometric properties of the kite's diagonals.

The diagonals

WY

and

XZ

are perpendicular at intersection point

M

, creating four right triangles inside the kite.

By definition, the diagonals of a kite are perpendicular to each other.

Calculate the lengths of the upper pair of equal sides (

WX

and

WZ

WX = \sqrt{WM^2 + XM^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}

. Since

WZ = WX

WZ = 10\text{ cm}

Apply the Pythagorean theorem to right triangle

WMX

with legs of length 6 cm and 8 cm.

Calculate the lengths of the lower pair of equal sides (

YX

and

YZ

YX = \sqrt{MY^2 + XM^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17\text{ cm}

. Since

YZ = YX

YZ = 17\text{ cm}

Apply the Pythagorean theorem to right triangle

YMX

with legs of length 15 cm and 8 cm.

Compute the total perimeter of kite

WXYZ

\text{Perimeter} = WX + WZ + YX + YZ = 10 + 10 + 17 + 17 = 54\text{ cm}

Sum the lengths of all four exterior sides.

Key Concept

Properties of Kite Diagonals and Pythagorean Theorem

Estimated Time:1m 15s

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