Question

1. match.

the 30°-60°-90° triangle theorem states that in a 30°-60°-90° triangle, the length of the hypotenuse is the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg.

a. 3 times
b. √3
c. twice
d. √2

1. choose the best answer.

a 45°-45°-90° right triangle is also called an ____ right triangle.

isosceles
acute

Explanation:

2)

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the shorter leg (opposite \(30^\circ\)) is \(1\) part, the longer leg (opposite \(60^\circ\)) is \(\sqrt{3}\) parts, and the hypotenuse (opposite \(90^\circ\)) is \(2\) parts.

Step2: Determine hypotenuse relation

The hypotenuse length is \(2\) times the shorter leg (since \(2 = 2\times1\)).

Step3: Determine longer leg relation

The longer leg length is \(\sqrt{3}\) times the shorter leg (since \(\sqrt{3}= \sqrt{3}\times1\)).

A \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle has two equal angles (\(45^\circ\) each), so the sides opposite these angles (the legs) are equal in length. A triangle with two equal sides is called an isosceles triangle. An acute triangle has all angles less than \(90^\circ\), but this triangle is right - angled (has a \(90^\circ\) angle), so "isosceles" is correct.

Answer:

First blank: c. twice; Second blank: b. \(\boldsymbol{\sqrt{3}}\)

3)