Question

19)
20)
21)
22)
23)
24)

Explanation:

Problem 19:

Step1: Identify triangle type (30-60-90)

In a 30-60-90 triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\) (opposite 30°, 60°, hypotenuse). The side opposite 30° is 2 (the shorter leg).

Step2: Find hypotenuse \(a\)

The hypotenuse is twice the shorter leg: \(a = 2 \times 2 = 4\)

Step3: Find longer leg \(b\)

The longer leg is shorter leg times \(\sqrt{3}\): \(b = 2 \times \sqrt{3} = 2\sqrt{3}\)

Step1: Identify triangle type (45-45-90)

In a 45-45-90 triangle, the legs are equal, and hypotenuse is leg \(\times \sqrt{2}\). Let leg length be \(l\), hypotenuse \(h = l\sqrt{2}\).

Step2: Solve for \(u\) and \(v\)

Given hypotenuse \(h = 5\), so \(l = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2}\). Thus, \(u = v = \frac{5\sqrt{2}}{2}\)

Step1: Identify triangle type (45-45-90)

Legs are equal, hypotenuse is leg \(\times \sqrt{2}\). One leg is 4, so the other leg \(y = 4\).

Step2: Find hypotenuse \(x\)

\(x = 4 \times \sqrt{2} = 4\sqrt{2}\)

Answer:

\(a = 4\), \(b = 2\sqrt{3}\)

Problem 20: