In right triangle ABC, the measure of angle B is 90^\circ. The acute angles A and C satisf… — SAT

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . The acute angles $A$ and $C$ satisfy $\sin(A) = \frac{k}{3}$ and $\cos(C) = \frac{4}{k+4}$ for some positive constant $k$ . What is the value of $k$ ?

A
6
B
8
2Answer
D
12

Answer

In right triangle

ABC

, the measure of angle

B

90^\circ

, which means the acute angles

A

and

C

are complementary. Therefore, the co-function identity states that

\sin(A) = \cos(C)

. Equating the given expressions yields

\frac{k}{3} = \frac{4}{k+4}

. Cross-multiplying results in

k(k+4) = 12

, which simplifies to

k^2 + 4k - 12 = 0

. Factoring this equation gives

(k+6)(k-2) = 0

. This quadratic has solutions

k = -6

and

k = 2

. Since

k

must be positive, the value of

k

2

Step-by-Step Solution

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

90^\circ

, the sum of angles

A

and

C

90^\circ

, which means they are complementary angles. By the co-function identity,

\sin(A) = \cos(C)

In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

Equate the expressions for

\sin(A)

and

\cos(C)

and set up the equation for

k

\frac{k}{3} = \frac{4}{k+4}

This sets up the algebraic relation to solve for

k

using the given trigonometric expressions.

Solve the algebraic equation for

k

Cross-multiplying gives

k(k+4) = 12

, which expands to

k^2 + 4k = 12

. Subtracting

12

from both sides results in the quadratic equation

k^2 + 4k - 12 = 0

. Factoring this equation yields

(k+6)(k-2) = 0

Cross-multiplication eliminates the denominators, converting the equation to a quadratic form that can be solved by factoring.

Determine the valid positive value for

k

The solutions to the equation

(k+6)(k-2) = 0

are

k = -6

and

k = 2

. Since

k

must be a positive constant, we select

k = 2

The problem specifies that

k

is a positive constant, so the negative solution is discarded.

Key Concept

Co-function identity relating sine and cosine of complementary angles

Estimated Time:1m 30s

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