Question

the figure is a kite. what is the length of the kite’s longer diagonal?
○ 5 units
○ 35 units
○ 40 units
○ 48 units

Explanation:

Step1: Analyze the kite's properties

A kite has two diagonals, one of which is the perpendicular bisector of the other. The shorter diagonal is split into two equal parts of 12 units each (from the figure: 12 and 12). We can use the Pythagorean theorem to find the lengths of the segments of the longer diagonal.

Step2: Find the length of the upper segment of the longer diagonal

For the upper triangle (with hypotenuse 13 and base 12), let the height (segment of the longer diagonal) be \( h_1 \). By the Pythagorean theorem \( a^2 + b^2 = c^2 \), we have \( h_1^2 + 12^2 = 13^2 \).
\[
h_1^2 = 13^2 - 12^2 = 169 - 144 = 25
\]
\[
h_1 = \sqrt{25} = 5
\]

Step3: Find the length of the lower segment of the longer diagonal

For the lower triangle (with hypotenuse 37 and base 12), let the height (segment of the longer diagonal) be \( h_2 \). Using the Pythagorean theorem: \( h_2^2 + 12^2 = 37^2 \).
\[
h_2^2 = 37^2 - 12^2 = 1369 - 144 = 1225
\]
\[
h_2 = \sqrt{1225} = 35
\]

Step4: Calculate the length of the longer diagonal

The longer diagonal is the sum of \( h_1 \) and \( h_2 \). So, \( \text{Longer diagonal} = h_1 + h_2 = 5 + 35 = 40 \) units.

Answer:

40 units