Question

Which equations could be used to solve for the unknown lengths of AABC? Check all that apply.

Answer 1

Given that ABC is a right triangle.

The measure of ∠A is 45° and AB = 9

We need to determine the equations that could be used to solve the unknown lengths of ΔABC

Option A: $\sin \left(45^{\circ}\right)=\frac{BC}{9}$

The length of BC can be determined using the trigonometric ratios.

$sin\ \theta=\frac{opp}{hyp}$

where $\theta=45^{\circ}$, $opp= BC$ and $hyp = 9$

Hence, substituting the values, we get;

$\sin \left(45^{\circ}\right)=\frac{BC}{9}$

Hence, Option A is the correct answer.

Option B: $\sin \left(45^{\circ}\right)=\frac{9}{BC}$

The length of BC can be determined using the trigonometric ratios.

$sin\ \theta=\frac{opp}{hyp}$

where $\theta=45^{\circ}$, $opp= BC$ and $hyp = 9$

Hence, substituting the values, we get;

$\sin \left(45^{\circ}\right)=\frac{BC}{9}$

Thus, the length of BC can be determined using $\sin \left(45^{\circ}\right)=\frac{BC}{9}$

Hence, Option B is not the correct answer.

Option C: $9 \tan \left(45^{\circ}\right)=A C$

The length of AC can be determined using the trigonometric ratios.

$tan \ \theta= \frac{opp}{adj}$

where $\theta=45^{\circ}$, $opp= BC$ and $adj=AC$

Substituting the values, we get;

$tan \ 45^{\circ}=\frac{BC}{AC}$

Thus, the length of AC using the trigonometric ratios is $tan \ 45^{\circ}=\frac{BC}{AC}$

Hence, Option C is not the correct answer.

Option D: $(A C) \sin \left(45^{\circ}\right)=B C$

The formula for $sin \ \theta$ is given by the formula,

$sin\ \theta=\frac{opp}{hyp}$

where $\theta=45^{\circ}$, $opp= BC$ and $hyp = 9$

Hence, substituting the values, we get;

$\sin \left(45^{\circ}\right)=\frac{BC}{9}$

Thus, the given equation $(A C) \sin \left(45^{\circ}\right)=B C$ is wrong.

Hence, Option D is not the correct answer.

Option E: $\cos \left(45^{\circ}\right)=\frac{BC}{9}$

The formula for $cos \ \theta$ is given by the formula,

$cos \ \theta=\frac{adj}{hyp}$

where $\theta=45^{\circ}$, $adj=AC$ and $hyp = 9$

Substituting the values, we get;

$\cos \left(45^{\circ}\right)=\frac{AC}{9}$

Hence, the given equation $\cos \left(45^{\circ}\right)=\frac{BC}{9}$ is not possible.

Thus, Option E is not the correct answer.