Question

In triangle ABC, angle ABC = 90º, and point D lies on segment BC such that AD is an angle bisector. If AB = 12 and BD = 4, then find AC

Answer 1

To find AC in triangle ABC, we can use the angle bisector theorem and the Pythagorean theorem. By setting up an equation using the angle bisector theorem and then using the Pythagorean theorem, we can solve for AC. AC is approximately 15.62.

Step-by-step explanation:

To find AC, we can use the angle bisector theorem and the Pythagorean theorem. Since AD is an angle bisector, we can use the angle bisector theorem to determine the ratio of BD to DC. Since BD = 4, we can let DC = x. Then, according to the angle bisector theorem, 4/x = 12/(AC+4). Cross-multiplying and simplifying, we get x = (AC + 4)/3.

Next, we can use the Pythagorean theorem to find the value of AC. In triangle ABC, we know AB = 12 and BC = BD + DC = 4 + x. Since angle ABC is a right angle, we can use the Pythagorean theorem: (AC+4)^2 = 12^2 + (4+x)^2. Simplifying and solving the equation, we get AC ≈ 15.62.

Answer 2

The value for the length of AC of triangle ∆ABC is equal to 15 to the nearest whole number.

The angle m∠ABC = 90° implies that triangle ABC and ABD are right triangles, which implies we can use the trigonometric ratios to evaluate for the angles and lengths as follows;

Considering the right triangle ∆ABD we have

tan BAD = 4/12 {opposite/adjacent}

tan BAD = 1/3

m∠BAD = tan⁻¹(1/3)

Thus for the right triangle ∆ABC since AD is an angle bisector then;

m∠BAC = 2tan⁻¹(1/3)

To get AC, we use the cosine of m∠BAC so that;

cos m∠BAC = 12/AC {adjacent/hypotenuse}

AC = 12/cos m∠BAC

AC = 12/ cos[2tan⁻¹(1/3)]

AC = 15 approximately to the nearest whole number.

Therefore, the length of AC which is the hypotenuse of the right triangle ∆ABC is equal to 15.