Select the correct answer.

The vertices of a triangle are [tex]A(7,5), B(4,2)[/tex], and [tex]C(9,2)[/tex]. What is [tex]m \angle ABC[/tex]?

A. [tex]30^{\circ}[/tex]
B. [tex]45^{\circ}[/tex]
C. [tex]56.31^{\circ}[/tex]
D. [tex]78.69^{\circ}[/tex]



Answer :

To find the measure of [tex]\(\angle ABC\)[/tex] in the triangle with vertices [tex]\(A(7,5)\)[/tex], [tex]\(B(4,2)\)[/tex], and [tex]\(C(9,2)\)[/tex], we need to follow these steps:

1. Determine the lengths of the sides of the triangle using the distance formula:
[tex]\[ AB = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2} \][/tex]
[tex]\[ BC = \sqrt{(C_x - B_x)^2 + (C_y - B_y)^2} \][/tex]
[tex]\[ AC = \sqrt{(C_x - A_x)^2 + (C_y - A_y)^2} \][/tex]

Using the vertices given:
[tex]\[ AB = \sqrt{(4 - 7)^2 + (2 - 5)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.242640687119285 \][/tex]

[tex]\[ BC = \sqrt{(9 - 4)^2 + (2 - 2)^2} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5 \][/tex]

[tex]\[ AC = \sqrt{(9 - 7)^2 + (2 - 5)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275463989 \][/tex]

2. Apply the Law of Cosines to find [tex]\(\cos \angle ABC\)[/tex]:
[tex]\[ \cos(\angle ABC) = \frac{AB^2 + BC^2 - AC^2}{2 \times AB \times BC} \][/tex]

Substituting the lengths we have calculated:
[tex]\[ \cos(\angle ABC) = \frac{(4.242640687119285)^2 + (5)^2 - (3.605551275463989)^2}{2 \times 4.242640687119285 \times 5} \][/tex]

[tex]\[ \cos(\angle ABC) = \frac{18 + 25 - 13}{2 \times 4.242640687119285 \times 5} = \frac{30}{42.42640687119285} \approx 0.7071067811865476 \][/tex]

3. Find the angle [tex]\( \angle ABC \)[/tex] using the arccosine function:
[tex]\[ \angle ABC = \arccos(0.7071067811865476) \][/tex]

[tex]\[ \angle ABC \approx 45.0^{\circ} \][/tex]

So, the measure of [tex]\(\angle ABC\)[/tex] is closest to [tex]\(45^{\circ}\)[/tex].

Therefore, the correct answer is [tex]\(B\)[/tex].