Question

fill in the blank 1 point if a=10, find b and c. b= choose your answer... c= choose your answer...

Explanation:

This is a 30 - 60 - 90 right triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest one (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step 1: Identify the sides

In the given triangle, side \(a\) is opposite the \(30^{\circ}\) angle? Wait, no. Wait, the right angle is between \(a\) and \(b\). The angle of \(30^{\circ}\) is at the end of \(b\), so the side opposite \(30^{\circ}\) is \(a\)? Wait, no. Let's look at the angles. The right angle is between \(a\) (vertical leg) and \(b\) (horizontal leg). The angle at the bottom right is \(30^{\circ}\), so the side opposite \(30^{\circ}\) is \(a\) (vertical leg). The side opposite \(60^{\circ}\) is \(b\) (horizontal leg), and the hypotenuse is \(c\).

So in a 30 - 60 - 90 triangle, if the side opposite \(30^{\circ}\) (let's call it \(x\)) is \(a = 10\), then:

• The side opposite \(60^{\circ}\) (which is \(b\)) is \(x\sqrt{3}\)
• The hypotenuse \(c\) is \(2x\)

Wait, no. Wait, maybe I mixed up. Let's re - establish:

In a right triangle, the side opposite \(30^{\circ}\) is the shorter leg. Let's check the angles. The angle at the bottom is \(30^{\circ}\), so the side opposite to it is \(a\) (the vertical leg). So \(a\) is opposite \(30^{\circ}\), so \(a=x\), \(b\) (opposite \(60^{\circ}\)) is \(x\sqrt{3}\), and \(c\) (hypotenuse) is \(2x\).

Given \(a = 10\) (so \(x = 10\))

Step 2: Find \(b\)

Since \(b\) is opposite \(60^{\circ}\), and in 30 - 60 - 90 triangle, the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), where \(x=a = 10\)

So \(b=x\sqrt{3}=10\sqrt{3}\)

Step 3: Find \(c\)

The hypotenuse \(c\) in a 30 - 60 - 90 triangle is \(2x\), where \(x = a=10\)

So \(c = 2x=2\times10 = 20\)

Answer:

\(b = 10\sqrt{3}\), \(c = 20\)