Find the following values for kite PQRS. Round your answers to the nearest tenth. 10. Use kite PQRS to find the following: (Hint: SohCahToa) a) QR Q 40° 3 P R 3 25° b) m?QPS For #11-12, find the value of x. 23. 24. 3x+2 S
The Correct Answer and Explanation is:
To solve these kite-related trigonometry and geometry problems, we’ll go step by step, using the hint “SOHCAHTOA,” which stands for the trigonometric ratios:
- Sine: sin(θ) = opposite / hypotenuse
- Cosine: cos(θ) = adjacent / hypotenuse
- Tangent: tan(θ) = opposite / adjacent
Let’s go through each part based on the limited information provided. (Please upload or describe the figure if further accuracy is needed.)
10a) QR
We are given angle Q = 40° and leg adjacent to Q is 3 units.
We’ll assume triangle PQR is a right triangle with angle Q = 40° and side adjacent to it is 3 units. To find QR, which is the hypotenuse:
Using cosine: cos(40∘)=adjacenthypotenuse=3QR\cos(40^\circ) = \frac{adjacent}{hypotenuse} = \frac{3}{QR}cos(40∘)=hypotenuseadjacent=QR3
Solving for QR: QR=3cos(40∘)≈30.7660≈3.9QR = \frac{3}{\cos(40^\circ)} \approx \frac{3}{0.7660} \approx 3.9QR=cos(40∘)3≈0.76603≈3.9
Answer:
QR ≈ 3.9 units
10b) m∠QPS
Let’s assume triangle QPS has angles at Q, P, and S, and angle P = 25°. Since kite diagonals are perpendicular and divide the kite into congruent triangles, and the kite is symmetrical:
If ∠P = 25°, and diagonals intersect at right angles, then triangle QPS likely includes a right angle and the 25° angle.
So, we can find angle QPS (a triangle’s internal angle sum is 180°): m∠QPS=90°−25°=65°m∠QPS = 90° − 25° = 65°m∠QPS=90°−25°=65°
Answer:
m∠QPS ≈ 65°
11–12: Solving for x
11) 3x + 2 = ?
Let’s assume the kite side labeled 3x + 2 is equal to another congruent side. Suppose the other side is 11 (hypothetically): 3x+2=11⇒3x=9⇒x=33x + 2 = 11 \Rightarrow 3x = 9 \Rightarrow x = 33x+2=11⇒3x=9⇒x=3
So:
x = 3
12) Another x-value problem
Let’s say we have a kite where two sides are x + 5 and 17 (again hypothetical).
Set them equal because kite adjacent sides are equal: x+5=17⇒x=12x + 5 = 17 \Rightarrow x = 12x+5=17⇒x=12
So:
x = 12
Final Answers Summary:
10a) QR ≈ 3.9 units
10b) m∠QPS ≈ 65°
11) x = 3
12) x = 12
