Question

a telephone pole broke into two parts, the upper part of the broken pole leans to touch the ground as pictured below.

annika recognizes that the situation approximates the shape of a right triangle. in which scenario is she correctly using properties of right triangles to learn more about the situation?

a. she can use the pythagorean theorem, the angle formed by the ground and the upper piece of the pole, and the height at which the pole broke to find the length of the upper piece of the pole.

b. she can use the tangent of the angle formed where the the upper piece of the pole meets the ground with the distance from that angle to the the base of the pole to find the height at which the pole broke.

c. she can use the pythagorean theorem, the height at which the pole broke, and the distance from the base of the pole to where the upper piece touches the ground to find the measure of the angle formed by the two pieces of the pole.

d. she can use the cosine of the angle formed where the upper piece of the pole meets the ground with the height at which the pole broke to find the length of the upper piece of the pole.

Explanation:

To solve this, we analyze each option based on right - triangle properties (trigonometric ratios: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ and Pythagorean theorem: $a^{2}+b^{2}=c^{2}$ where $c$ is hypotenuse, $a,b$ are legs).

Step 1: Analyze Option A

The Pythagorean theorem requires two sides (legs or leg and hypotenuse) to find the third. If we know an angle (not a right angle) and a leg (height at which pole broke), we can use trigonometric ratios, not just Pythagorean theorem with an angle. So A is incorrect.

Step 2: Analyze Option B

Let the height at which the pole broke be $a$ (opposite side), the distance from the angle to the base be $b$ (adjacent side), and the angle be $\theta$. $\tan\theta=\frac{a}{b}\implies a = b\times\tan\theta$. If we know the angle and the adjacent side ($b$), we can find the opposite side ($a$), which is the height at which the pole broke. This uses the tangent ratio correctly.

Step 3: Analyze Option C

The Pythagorean theorem relates the sides, not the angles. To find an angle between the two pieces of the pole (a non - right angle), we would use trigonometric ratios (like inverse trigonometric functions) with the sides, not the Pythagorean theorem. So C is incorrect.

Step 4: Analyze Option D

Let the height at which the pole broke be $a$ (opposite side), the length of the upper piece be $c$ (hypotenuse), and the angle be $\theta$. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, but the height at which the pole broke is the opposite side, not the adjacent side. So $\cos\theta$ is not related to the height (opposite) and hypotenuse in the way described. So D is incorrect.

Answer:

B. She can use the tangent of the angle formed where the the upper piece of the pole meets the ground with the distance from that angle to the the base of the pole to find the height at which the pole broke.