Question

In triangle, a, b, c, comma△abc, angle, a, \cong, angle, c, comma∠a≅∠c, b, c, equals, 12bc=12 and c, a, equals, 10ca=10. find a, b, .ab.

a) 8
b) 12
c) 15
d) 20

Answer 1

Using the Law of Cosines, we can calculate the length of side AB in triangle ABC, which is an isosceles triangle with sides BC = 12 and CA = 10. By first finding the measure of angle B and then applying the Law of Cosines formula, we resolve for side AB.

Step-by-step explanation:

The student has described a triangle ABC where angle A is congruent to angle C, side BC equals 12, and side CA equals 10. To find side AB, we can use the Law of Cosines since we have two sides and the included angle (which is the angle opposite to the side we are trying to find).

Let's denote the length of side AB with x. In a triangle where the angles at A and C are congruent, this means triangle ABC is an isosceles triangle with angle B as the apex angle. Since angles A and C are congruent, and the sum of all angles in a triangle is 180 degrees, we can find angle B by calculating 180 degrees minus 2 times angle A (or angle C).

After finding angle B, we can apply the Law of Cosines:

c2 = a2 + b2 - 2ab cos(y)

where:

• c is the side opposite the angle (AB or x in our case)
• a and b are the known sides (AC = 10 and BC = 12)
• y is the included angle (angle B)

Plugging in the known values, we solve for x to find the length of side AB.