In triangular plot ABC, the boundary lengths are AB = 13 meters, BC = 8 meters, and AC = 1… — ACT

In triangular plot $ABC$ , the boundary lengths are $AB = 13$ meters, $BC = 8$ meters, and $AC = 15$ meters. A straight drainage pipe is laid from vertex $B$ perpendicular to side $AC$ , meeting side $AC$ at point $D$ . What is the distance, in meters, from point $A$ to point $D$ ?

Answer: 11 meters

Answer

The distance from point A to point D is 11 meters.

Applying the Law of Cosines

a^2 = b^2 + c^2 - 2bc \cos A

with

a=8

b=15

, and

c=13

yields

64 = 225 + 169 - 390 \cos A

, which simplifies to

390 \cos A = 330

, giving

\cos A = \frac{11}{13}

. In right triangle

ABD

\cos A = \frac{AD}{AB}

, so

AD = 13 \cdot \frac{11}{13} = 11

meters.

Step-by-Step Solution

Apply the Law of Cosines to triangle ABC to solve for the cosine of angle A.

cos A = 11/13

The Law of Cosines relates all three side lengths of a triangle to the cosine of one of its interior angles.

Use right triangle trigonometry in right triangle ABD to calculate the length of AD.

AD = 11 meters

Since BD is perpendicular to AC, triangle ABD is a right triangle with hypotenuse AB and adjacent side AD relative to angle A.

Key Concept

Law of Cosines and Right Triangle Trigonometry

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic