Question

i 177b (4) (7) (5) (8) (6) (9)

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, the sides are in the ratio $1:\sqrt{3}:2$. If the side opposite the 30 - degree angle is $a$, the side opposite the 60 - degree angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Solve for (4)

The side opposite the 60 - degree angle is $5\sqrt{3}$. Let the side opposite the 30 - degree angle be $a$. Since the side opposite the 60 - degree angle is $a\sqrt{3}$, then $a\sqrt{3}=5\sqrt{3}$, so $a = 5$. The hypotenuse $x=2a = 10$.

Step3: Solve for (5)

The hypotenuse is 6. Since the hypotenuse in a 30 - 60 - 90 triangle is $2a$, then $2a = 6$, so $a = 3$. The side opposite the 30 - degree angle $x=a = 3$.

Step4: Solve for (6)

The side opposite the 60 - degree angle is $3\sqrt{3}$. Since the side opposite the 60 - degree angle is $a\sqrt{3}$, then $a\sqrt{3}=3\sqrt{3}$, so $a = 3$. The side opposite the 30 - degree angle $x=a = 3$.

Step5: Solve for (7)

The side opposite the 30 - degree angle is 3. Since the hypotenuse in a 30 - 60 - 90 triangle is $2a$, and here $a = 3$, the hypotenuse $x=2a=6$.

Step6: Solve for (8)

The side opposite the 60 - degree angle is 2. Since the side opposite the 60 - degree angle is $a\sqrt{3}$, then $a\sqrt{3}=2$, so $a=\frac{2}{\sqrt{3}}$. The hypotenuse $x = 2a=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3}$.

Step7: Solve for (9)

The side opposite the 30 - degree angle is $3\sqrt{6}$. Since the side opposite the 60 - degree angle $x$ in a 30 - 60 - 90 triangle is $a\sqrt{3}$ and the side opposite 30 - degree angle is $a$, here $a = 3\sqrt{6}$, so $x=a\sqrt{3}=3\sqrt{6}\times\sqrt{3}=9\sqrt{2}$.

Answer:

(4) $x = 10$
(5) $x = 3$
(6) $x = 3$
(7) $x = 6$
(8) $x=\frac{4\sqrt{3}}{3}$
(9) $x = 9\sqrt{2}$