Question

A lamppost, CAB, bent at point A after a storm. The tip of the lamppost touched the ground at point C, as shown below: Triangle ABC has measure of angle C equal to 45 degrees, measure of angle ABC equal to 90 degrees, and length of BC equal to 12 feet. What is the height, in feet, of the portion AB of the lamppost?

Answer 1

The height of the portion AB of the lamppost is 12 feet.

Step-by-step explanation:

To find the height of the portion AB of the lamppost, we can use trigonometry and the given angles and side lengths of triangle ABC.

Since angle C is 45 degrees and angle ABC is 90 degrees, angle BAC can be found by subtracting the sum of angles C and ABC from 180 degrees: 180 - 45 - 90 = 45 degrees.

Now, we can use the tangent function to find the height AB: tan(BAC) = AB / BC. Substituting the known values, tan(45) = AB / 12. Solving for AB, we get AB = 12 feet.

Answer 2

To find the height of the portion AB of the lamppost we can use basic trigonometry. Since we know the length of BC and the measure of angle ABC we can use the tangent function.

In triangle ABC the tangent of angle ABC is defined as the ratio of the length of the side opposite the angle (AB) to the length of the side adjacent to the angle (BC).

So we have:

tan(ABC) = AB / BC

Since the measure of angle ABC is 90 degrees the tangent of 90 degrees is undefined. However we can consider the acute angle formed between AB and BC which is the complement of angle ABC. In this case the acute angle is 90 - 45 = 45 degrees.

So we have:

tan(45) = AB / BC

Since the tangent of 45 degrees is equal to 1 we can solve for AB:

1 = AB / 12

Cross-multiplying we get:

AB = 12 feet

Therefore the height of the portion AB of the lamppost is 12 feet.