Question

In triangle ABC, angle A is 30°, and angle C is a right angle. The length of side AB is 3. Find the lengths of the other sides: BC and AC. Give the exact answers, but you may not use 'sin', 'cos', or 'tan'. Use '√(3)' or '3^(.5)' for root.

a. True
b. False

Answer 1

The lengths of the sides AC and BC in the right triange ABC, where angle A is 30° and angle C is 90°, are 1.5 and 1.5√3 respectively, using the Pythagorean theorem.

Step-by-step explanation:

To find the lengths of the other sides of the triangle ABC, we can apply the Pythagorean theorem. Since angle C is a right angle, side AC and BC are the legs, and AB is the hypotenuse. The Pythagorean theorem states that the square of the hypotenuse (AB) is equal to the sum of the squares of the legs (AC and BC). Therefore:

AB² = AC² + BC²

Since AB is given as 3, and ACA is 30° in a right-angled triangle, we know AC is half of AB in a 30°-60°-90° triangle:

AC = AB/2 = 3/2 = 1.5

Now to find BC, we use the Pythagorean theorem:

3² = (1.5)² + BC²

9 = 2.25 + BC²

BC² = 9 - 2.25

BC² = 6.75

BC = √6.75

To express √6.75 in terms of √3:

BC = √(2.25 ⋅ 3) = √(1.5² ⋅ 3) = 1.5√3