Question

priya says, \i know everything about a right triangle with a 30 degree angle and a hypotenuse with length 1 cm. here, 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: ° and °
missing side lengths: cm and cm
explain or show your reasoning.

Explanation:

Step1: Determine Angles

In a right triangle, angles sum to \(180^\circ\). One angle is \(90^\circ\), another is \(30^\circ\), so the third angle is \(180 - 90 - 30 = 60^\circ\). Scaling the hypotenuse doesn't change angles (similar triangles), so angles remain \(90^\circ\) and \(60^\circ\).

Step2: Determine Side Lengths

For a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, sides are in ratio \(1 : \sqrt{3} : 2\) (opposite \(30^\circ\), \(60^\circ\), hypotenuse). Original hypotenuse \(1\) cm: opposite \(30^\circ\) is \(0.5\) cm, adjacent \(30^\circ\) is \(\approx 0.866\) cm. New hypotenuse \(2\) cm (scaled by 2). So:

• Side opposite \(30^\circ\): \(0.5 \times 2 = 1\) cm.
• Side adjacent \(30^\circ\) (opposite \(60^\circ\)): \(0.866 \times 2 \approx 1.732\) cm (or \(\sqrt{3}\) cm).

Answer:

Missing angles: \(90^\circ\) and \(60^\circ\)
Missing side lengths: \(1\) cm and \(\approx 1.732\) (or \(\sqrt{3}\)) cm