Question

In triangle ABC, AB = 6 cm, AC = BC = 7 cm. Also, CP is a median. Which of the following is the area of triangle ABC?

• A. \( \frac{2}{\sqrt{10}} \, \text{cm}^2 \)
• B. \( \frac{6}{\sqrt{3}} \, \text{cm}^2 \)
• C. \( \frac{8}{\sqrt{3}} \, \text{cm}^2 \)
• D. \( \frac{12}{\sqrt{2}} \, \text{cm}^2 \)
• E. \( \frac{6}{\sqrt{10}} \, \text{cm}^2 \)

Hint:

For triangles with a median, the area can be found using Heron’s formula or by using the property of the median dividing the area into two equal parts.

Answer 1

The Correct Option is A

Step 1: In triangle ABC, we are given that AB = 6 cm, AC = BC = 7 cm, and CP is a median. Since CP is a median, it divides the triangle into two triangles of equal area. We need to find the area of triangle ABC.

Step 2: We will use Heron’s formula to find the area of triangle ABC. The semi-perimeter \(s\) of triangle ABC is given by: \[ s = \frac{AB + AC + BC}{2} = \frac{6 + 7 + 7}{2} = 10 \] Step 3: Using Heron’s formula, the area \(A\) of triangle ABC is: \[ A = \sqrt{s(s - AB)(s - AC)(s - BC)} = \sqrt{10(10 - 6)(10 - 7)(10 - 7)} = \sqrt{10 \times 4 \times 3 \times 3} = \sqrt{360} = 6\sqrt{10} \, \text{cm}^2 \] Step 4: Thus, the area of triangle ABC is \( 6\sqrt{10} \, \text{cm}^2 \). Therefore, the correct answer is \( \boxed{(1)} \) \( \frac{2}{\sqrt{10}} \, \text{cm}^2 \).