Question

quiz - special triangles
1
triangle with right angle, 60° angle, hypotenuse 7, sides m and n
a) $m = \frac{7\sqrt{3}}{2}$, $n = \frac{7\sqrt{3}}{2}$
b) $m = 7\sqrt{3}$, $n = \frac{7}{2}$
c) $m = \frac{7}{2}$, $n = 7\sqrt{3}$
d) $m = \frac{7\sqrt{3}}{2}$, $n = \frac{7}{2}$
\bigcirc a
\bigcirc b
\bigcirc c
\bigcirc d

Explanation:

Step1: Identify Triangle Type

The triangle is a right - triangle (one right angle) with a \(60^{\circ}\) angle. So the third angle is \(180 - 90 - 60=30^{\circ}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Here, the hypotenuse is \(7\). Let the side opposite \(30^{\circ}\) be \(n\) and the side opposite \(60^{\circ}\) be \(m\).

Step2: Find the Shortest Side (\(n\))

Since the hypotenuse \(= 2x\) and hypotenuse \( = 7\), then \(2x=7\), so \(x=\frac{7}{2}\). The side \(n\) is opposite the \(30^{\circ}\) angle, so \(n = x=\frac{7}{2}\).

Step3: Find the Side Opposite \(60^{\circ}\) (\(m\))

The side \(m\) is opposite the \(60^{\circ}\) angle, so \(m=x\sqrt{3}\). Substituting \(x = \frac{7}{2}\), we get \(m=\frac{7\sqrt{3}}{2}\).

Answer:

D. \(m=\frac{7\sqrt{3}}{2},n = \frac{7}{2}\)