Question

If rectangle \(A B C D\) has an area of 48 and the tangent of \(\angle B C A\) (not shown) is \(\frac{3}{4}\), then which of the following is the length of \(\overline{B D}\) (not shown)? A) 5 B) 10 C) 13 D) It cannot be determined from the given information.

Short answer

The length of diagonal \(\overline{BD}\) is 10. The answer is B) 10.

Step-by-step solution

Area of the rectangle

The area of the rectangle is given to be 48. Let the side AD have a length of x and the side BC have a length of y. The area of the rectangle can be calculated as follows: \(x \cdot y = 48\).

Using Trigonometry

It is given that \(\tan(\angle BCA) =\frac{3}{4}\). Since this angle is formed by the sides BC and AD, we can write the tangent as follows: \(\tan(\angle BCA) =\frac{AD}{BC}=\frac{x}{y}\). Now, we have two equations: 1. \(x \cdot y = 48\) 2. \(\frac{x}{y} =\frac{3}{4}\)

Solving for the length of side AD

To solve for the length of side AD (x), we can use these equations by solving Eq(2) for y and substituting it in Eq(1). From Eq(2), we have: \(y=\frac{4x}{3}\) Now, substitute this into Eq(1): \(x \cdot \frac{4x}{3} = 48\) Multiply both sides by 3: \(4x^2 = 144\) Divide both sides by 4: \(x^2 = 36\) Taking the square root of both sides, we get: \(x = 6\) Now that we have the length of AD (x), we can find the length of BC.

Solving for the length of side BC

We can use the formula we derived in Step 2 to find the length of BC (y): \(y = \frac{4x}{3}\) Substitute x = 6: \(y = \frac{4 \cdot 6}{3}\) \(y = \frac{24}{3}\) \(y = 8\)

Finding the length of BD using the Pythagorean theorem

Now that we have the lengths of AD and BC, we can use the Pythagorean theorem to find the length of the diagonal BD. Using the Pythagorean theorem, we have: \(BD^2 = AD^2 + BC^2\) \(BD^2 = 6^2 + 8^2\) \(BD^2 = 36 + 64\) \(BD^2 = 100\) Taking the square root of both sides, we get: \(BD = 10\) Thus, the length of diagonal \(\overline{BD}\) is 10. The answer is B) 10.