Question

Find the perimeter of Isosceles triangle ABC (below) if mAD = 3 and m\(\angle\)BAC = 55 degrees. Round to the nearest hundredth.

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• A. 5.21
• B. 10.42
• C. 13.48
• D. 16.46
• E. 13.39

Hint:

For isosceles triangle problems, dropping an altitude from the vertex angle is a very common and useful strategy. It creates two congruent right-angled triangles, allowing you to use the Pythagorean theorem and trigonometric ratios.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem asks for the perimeter of an isosceles triangle. The perimeter is the sum of the lengths of all three sides. We are given the length of a segment of the base and a base angle. We will use properties of isosceles triangles and trigonometry to find the lengths of the sides.

Step 2: Key Formula or Approach:
- Perimeter \( P = AB + BC + AC \).
- In an isosceles triangle ABC with vertex B, \( AB = BC \).
- The altitude from the vertex angle to the base (BD) bisects the base. So, \( AC = 2 \times AD \).
- Basic trigonometric ratio (SOH-CAH-TOA): In a right-angled triangle, \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

Step 3: Detailed Explanation:
The triangle ABC is isosceles. The altitude BD is drawn from the vertex B to the base AC. In an isosceles triangle, this altitude is also the median, so it bisects the base AC at point D.
We are given that the length of segment AD is 3.
\[ AD = 3 \] Therefore, the length of the full base AC is:
\[ AC = 2 \times AD = 2 \times 3 = 6 \] Now we need to find the length of the equal sides, AB and BC. Let's focus on the right-angled triangle \(\triangle\)ABD.
We are given \( \angle BAC \) (which is the same as \( \angle BAD \)) is 55 degrees.
We know the length of the side adjacent to this angle, \( AD = 3 \).
We want to find the length of the hypotenuse, AB.
Using the cosine ratio:
\[ \cos(\angle BAD) = \frac{AD}{AB} \] \[ \cos(55^\circ) = \frac{3}{AB} \] Rearrange the formula to solve for AB:
\[ AB = \frac{3}{\cos(55^\circ)} \] Using a calculator, find the value of \( \cos(55^\circ) \).
\[ \cos(55^\circ) \approx 0.573576 \] \[ AB \approx \frac{3}{0.573576} \approx 5.231 \] Since triangle ABC is isosceles, \( BC = AB \approx 5.231 \).
Finally, calculate the perimeter of \(\triangle\)ABC.
\[ P = AB + BC + AC \] \[ P \approx 5.231 + 5.231 + 6 \] \[ P \approx 10.462 + 6 \] \[ P \approx 16.462 \] Rounding to the nearest hundredth, we get 16.46.

Step 4: Final Answer:
The perimeter of the isosceles triangle is approximately 16.46.