7. In a kite, the diagonals cross perpendicularly. One set of opposite angles is the same, and their diagonal is bisected by the other diagonal. Given a kite which has opposite angles of $30^{\circ}$ and $50^{\circ}$, and the diagonal not connected to these angles has a length of 10 inches, find the kite's area to the nearest hundredth.
Answer

Step 1 ：First, find the other two angles of the kite: 360° - 30° - 50° = 280°, so each of the other two angles is 140°.

Step 2 ：Divide the kite into two triangles by drawing the diagonal connected to the 30° and 50° angles. Let the length of this diagonal be d, and the length of the equal sides of the triangles be x.

Step 3 ：Use the sine rule to find x: \(x = \frac{\sin(140°) * d}{\sin(30°)}\)

Step 4 ：Use the Pythagorean theorem to find d: \(\left(\frac{x}{2}\right)^2 + 10^2 = d^2\)

Step 5 ：Substitute the expression for x into the equation: \(\left(\frac{\sin(140°) * d}{2 * \sin(30°)}\right)^2 + 10^2 = d^2\)

Step 6 ：Solve for d: \(d \approx 12.86\)

Step 7 ：Find x using the sine rule: \(x \approx 16.28\)

Step 8 ：Find the area of the two triangles: \(\text{Area}_1 = \frac{1}{2} * d * (x * \sin(30°)) \approx 64.28\) and \(\text{Area}_2 = \frac{1}{2} * d * (x * \sin(50°)) \approx 98.48\)

Step 9 ：Add the areas of the two triangles to find the kite's area: \(\text{Area} = \text{Area}_1 + \text{Area}_2 \approx 162.76\)

Step 10 ：\(\boxed{\text{The kite's area to the nearest hundredth is approximately 84.43 square inches}}\)