Question

find the first mistake in identifying the side opposite the 30° angle when the hypotenuse is 9.

1. ratio of sides: 1 : √3 : 2.
2. the shorter leg equals hypotenuse × √3/2.
3. compute 9 × 0.866 = 7.8.
4. conclude side opposite 30° = 7.8.

line 3
line 2
line 4
no mistakes were made.
line 1

Explanation:

Step1: Recall the side - ratio of a 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\), where the side opposite the \(30^{\circ}\) angle is the shortest side (let's call it \(a\)), the side opposite the \(60^{\circ}\) angle is \(a\sqrt{3}\), and the hypotenuse (\(c\)) is \(2a\). So, \(a=\frac{c}{2}\), and the side opposite the \(60^{\circ}\) angle is \(c\times\frac{\sqrt{3}}{2}\).

Step2: Analyze Line 2

Line 2 says "The shorter leg equals hypotenuse \(\times\frac{\sqrt{3}}{2}\)". But in a 30 - 60 - 90 triangle, the shorter leg (side opposite \(30^{\circ}\)) is \(\frac{\text{hypotenuse}}{2}\), and the longer leg (side opposite \(60^{\circ}\)) is \(\text{hypotenuse}\times\frac{\sqrt{3}}{2}\).

Answer:

Line 2