Question

select the correct answer.
the vertices of a triangle are a(7, 5), b(4, 2), and c(9, 2). what is m∠abc?
a. 30°
b. 45°
c. 56.31°
d. 78.69°

Explanation:

Step1: Calculate the lengths of sides

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
$AB=\sqrt{(7 - 4)^2+(5 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$
$BC=\sqrt{(9 - 4)^2+(2 - 2)^2}=\sqrt{25}=5$
$AC=\sqrt{(9 - 7)^2+(2 - 5)^2}=\sqrt{4 + 9}=\sqrt{13}$

Step2: Use the cosine - law

The cosine - law formula is $\cos B=\frac{AB^{2}+BC^{2}-AC^{2}}{2\cdot AB\cdot BC}$.
$AB^{2}=(3\sqrt{2})^{2}=18$, $BC^{2}=25$, $AC^{2}=13$.
$\cos B=\frac{18 + 25-13}{2\times3\sqrt{2}\times5}=\frac{30}{30\sqrt{2}}=\frac{\sqrt{2}}{2}$

Step3: Find the angle

Since $\cos B=\frac{\sqrt{2}}{2}$, then $B = 45^{\circ}$ (because $\cos45^{\circ}=\frac{\sqrt{2}}{2}$)

Answer:

B. $45^{\circ}$