Question

triangle mnp has vertices n(2,2) and p(0,-2). the triangle is symmetric about the y - axis. what is the approximate measure of the largest angle in the triangle? 52.8° 60.0° 63.6° 82.9°

Explanation:

Step1: Find the coordinates of point \(M\)

Since the triangle is symmetric about the \(y -\)axis, if \(N(2,2)\), then \(M(- 2,2)\) (because for a point \((x,y)\) symmetric about the \(y -\)axis, the symmetric point is \((-x,y)\)). And \(P(0,-2)\)

Step2: Calculate the lengths of the sides using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• Length of \(MN\): \(d_{MN}=\sqrt{(2 + 2)^2+(2 - 2)^2}=\sqrt{16}=4\)
• Length of \(MP\): \(d_{MP}=\sqrt{(-2-0)^2+(2 + 2)^2}=\sqrt{4 + 16}=\sqrt{20}\approx4.47\)
• Length of \(NP\): \(d_{NP}=\sqrt{(2-0)^2+(2 + 2)^2}=\sqrt{4 + 16}=\sqrt{20}\approx4.47\)

Step3: Use the Law of Cosines \(c^{2}=a^{2}+b^{2}-2ab\cos C\)

Let \(a = 4\), \(b = 4.47\), \(c = 4.47\). We want to find the angle opposite the longest side (but in this case \(MP = NP\)). Let's find the angle at \(M\).

$$
LATEXBLOCK0
$$

\(M=\cos^{-1}(0.447)\approx63.6^{\circ}\)

Answer:

\(63.6^{\circ}\)