Question

quiz 1: trigonometric ratios, identities, and functions
which of the following choices are the values of the side lengths of a triangle with 30°, 60°, 90° angles and a short leg = 13?
13, 13√3, 26
13, 13, 13√2
(13√3)/3, (13√3)/3, (26√3)/3
(13√2)/3, (13√2)/3, (26√2)/3

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the side ratios are \(1 : \sqrt{3} : 2\), where the short leg (opposite \(30^\circ\)) is \(1\), the long leg (opposite \(60^\circ\)) is \(\sqrt{3}\), and the hypotenuse is \(2\).

Step2: Apply given short leg length

Given short leg \(= 13\) (corresponds to the ratio \(1\)).

• Short leg: \(13\) (matches the ratio \(1\)).
• Long leg: \(13 \times \sqrt{3} = 13\sqrt{3}\) (matches the ratio \(\sqrt{3}\)).
• Hypotenuse: \(13 \times 2 = 26\) (matches the ratio \(2\)).

Check other options:

• Second option: Ratios \(1:1:\sqrt{2}\) (isosceles right triangle, \(45^\circ\)-\(45^\circ\)-\(90^\circ\)), incorrect.
• Third and fourth options: Ratios do not match \(30\)-\(60\)-\(90\) triangle ratios.

Answer:

A. \(13, 13\sqrt{3}, 26\) (assuming the first option is labeled A, with the text \(13, 13\sqrt{3}, 26\))