\(\textbf{ Find the Area of the Kite}\) Let \(d_1 = 16 \, \text{cm}\) and \(d_2 = 12 \, \text{cm}\)
\begin{align*}
\textbf{Area} = \amp \frac{1}{2} \times d_1 \times d_2
\end{align*}
\(\text{substitute} \, d_1 = 16 \, \text{cm and } d_2 = 12 \, \text{cm}\)
\begin{align*}
\text{Area} = \amp \frac{1}{2} \times 16 \, \text{cm } \times 12 \, \text{cm} \\
= \amp \frac{192}{2} \, \text{cm}^2 \\
= \amp 96 \, \text{cm}^2
\end{align*}
(b) Find the Perimeter of the Kite. The diagonals bisect each other at right angles, so each half-diagonal forms a right-angled triangle. \(\frac{d_1}{2} = 6\text { cm }\) and \(\frac{d_2}{2} = 8 \text{cm}\) Using the Pythagoras Theorem:
\begin{align*}
\amp a^2 + b^2 = c^2\\
= \amp 6^2 + 8^2\\
c^2 = \amp 36 + 64\\
\sqrt{c^2} = \amp \sqrt{100} \\
c = \amp 10 \, \text{cm}
\end{align*}
Since a kite has two pairs of equal sides, the perimeter is:
\begin{align*}
P = \amp 2(a+b) \\
= \amp 2(10 + 10) \, \text{cm}\\
= \amp 40 \, \text{cm}
\end{align*}
(c) Finding the Angles Using Trigonometry. Using trigonometry in the right-angled triangle:
\begin{align*}
\text{tan} \, \theta = \amp \frac{\text{Opposite}}{\text{adjacent}} \\
= \amp \frac{6}{8} \\
\theta = \amp \text{tan}^{-1} \, 0.75 \\
\amp \\
\theta = \amp 36.87^\circ
\end{align*}
A kite is a quadrilateral, meaning it has four angles.
The sum of the interior angles of any quadrilateral is given by the formula:
Sum of Interior Angles=
\((n-2) \times 180^\circ\)
where
\(n = 4\)(since a kite has 4 sides) \((4-2) \times 180^\circ \) \(= 2 \times 180^\circ\) \(= 360^\circ\)
Since the kite is symmetric, the larger angle ( \(\alpha \) )is
\begin{align*}
= \amp 360^\circ - 2 \theta \\
= \amp 360^\circ - 2 (36.87)^\circ \\
= \amp 360^\circ - 73.74 ^\circ \\
2(\alpha)= \amp 286.26 ^\circ\\
\alpha = \amp 143.13^\circ
\end{align*}
The angles of the kite are:
\(143^\circ, \, 143.13^\circ, \, 36.87^\circ \, \text{and} \, 36.87^\circ\)
