Question

7)
a) $x = 9$, $y = \frac{27}{4}$
b) $x = 9$, $y = \frac{9\sqrt{3}}{2}$
c) $x = \frac{9\sqrt{3}}{2}$, $y = \frac{27}{4}$
d) $x = \frac{9\sqrt{3}}{2}$, $y = \frac{9\sqrt{3}}{2}$
8)
a) $x = 7$, $y = 7$
b) $x = \frac{14\sqrt{3}}{3}$, $y = \frac{14\sqrt{3}}{3}$
c) $x = \frac{7\sqrt{3}}{2}$, $y = \frac{7\sqrt{2}}{2}$
d) $x = \frac{7\sqrt{2}}{2}$, $y = \frac{14\sqrt{3}}{3}$
9)
a) $x = \frac{5\sqrt{6}}{2}$, $y = 5$
b) $x = 5\sqrt{2}$, $y = \frac{5\sqrt{3}}{2}$
c) $x = \frac{5\sqrt{6}}{2}$, $y = \frac{5\sqrt{3}}{2}$
d) $x = 5\sqrt{2}$, $y = 5$
10)
a) $x = \frac{14\sqrt{3}}{3}$, $y = \frac{7\sqrt{3}}{3}$
b) $x = \frac{28\sqrt{3}}{3}$, $y = \frac{14\sqrt{3}}{3}$
c) $x = \frac{7\sqrt{3}}{3}$, $y = \frac{14\sqrt{3}}{3}$
d) $x = \frac{7\sqrt{3}}{3}$, $y = \frac{28\sqrt{3}}{3}$
11)
a) $18\sqrt{6}$
b) $36\sqrt{3}$
c) $9\sqrt{6}$
d) $6\sqrt{3}$
12)
a) $\frac{8\sqrt{30}}{3}$
b) $32\sqrt{5}$
c) $16\sqrt{10}$
d) $16\sqrt{5}$

Explanation:

Step1: Analyze the 30 - 60 - 90 triangle (Problem 7)

In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). Given the side opposite 30° is \(\frac{9}{2}\), the side opposite 60° (\(y\)) is \(\frac{9}{2}\times\sqrt{3}=\frac{9\sqrt{3}}{2}\), and the hypotenuse (\(x\)) is \(2\times\frac{9}{2} = 9\).

Step2: Analyze the 45 - 45 - 90 triangle (Problem 8)

In a 45 - 45 - 90 triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Given the hypotenuse is \(7\sqrt{2}\), then each leg (\(x,y\)) is \(\frac{7\sqrt{2}}{\sqrt{2}}=7\).

Step3: Analyze the 45 - 45 - 90 triangle (Problem 9)

In a 45 - 45 - 90 triangle, if one leg is 5, the hypotenuse (\(x\)) is \(5\sqrt{2}\), and the other leg (\(y = 5\)).

Step4: Analyze the 30 - 60 - 90 triangle (Problem 10)

In a 30 - 60 - 90 triangle, if the side opposite 30° (\(y\)) is \(\frac{7}{\sqrt{3}}=\frac{7\sqrt{3}}{3}\), the hypotenuse (\(x\)) is \(\frac{14\sqrt{3}}{3}\).

Step5: Analyze the combined triangle (Problem 11)

First, in the 60 - 90 triangle, the side adjacent to 60° is \(\frac{9\sqrt{3}}{\sqrt{3}} = 9\), the hypotenuse is \(18\). Then in the 45 - 90 triangle, \(x=\frac{18}{\sqrt{2}}=9\sqrt{2}\times\sqrt{3}=9\sqrt{6}\).

Step6: Analyze the combined triangle (Problem 12)

In the 60 - 90 triangle, the hypotenuse is \(16\sqrt{5}\). In the 45 - 90 triangle, \(x = 16\sqrt{5}\).

Answer:

1. B. \(x = 9,y=\frac{9\sqrt{3}}{2}\)
2. A. \(x = 7,y = 7\)
3. D. \(x = 5\sqrt{2},y = 5\)
4. A. \(x=\frac{14\sqrt{3}}{3},y=\frac{7\sqrt{3}}{3}\)
5. C. \(9\sqrt{6}\)
6. D. \(16\sqrt{5}\)