Question

priya says, \i know everything about a right triangle with a 30 degree angle and a hypotenuse with length 1 cm. h look.\

• the other angle is 60 degrees.
• the leg adjacent to the 30 degree angle is 0.866 cm long.
• the side opposite the 30 degree angle is 0.5 cm long.

han asks, \what would happen if a right triangle with a 30 degree angle has a hypotenuse that is 2 cm instead?\
help them find the missing angles and side lengths in the new triangle.
type your answers in the boxes.
missing angles: (60)° and (90)°
missing side lengths: (1.732) cm and (1) cm

Explanation:

Step1: Determine Angles

In a right triangle, the sum of angles is \(180^\circ\). One angle is \(90^\circ\) (right angle), one is \(30^\circ\). So the third angle is \(180 - 90 - 30 = 60^\circ\). Angles remain \(60^\circ\) and \(90^\circ\) (and \(30^\circ\)) as angle measures don't change with side length scaling.

Step2: Analyze Side Lengths (Scaling)

The original hypotenuse is \(1\) cm, new is \(2\) cm (scale factor \(k = 2\)).

• Side opposite \(30^\circ\) (original: \(0.5\) cm): New length \(= 0.5 \times 2 = 1\) cm.
• Leg adjacent to \(30^\circ\) (original: \(0.866\) cm, which is \(\frac{\sqrt{3}}{2} \approx 0.866\)): New length \(= \frac{\sqrt{3}}{2} \times 2 = \sqrt{3} \approx 1.732\) cm.

Answer:

Missing angles: \(60^\circ\) and \(90^\circ\)
Missing side lengths: \(1.732\) cm (adjacent to \(30^\circ\)) and \(1\) cm (opposite to \(30^\circ\))