Consider the following statements:<br>I. The line segment AP divides the area of the triangle ABC into two equal parts<br>II. The perimeter of the triangle APC is more than 46 cm<br>III. The area of the triangle APC is 50 square cm<br>Which of the statements given above are correct?

Let ABC be a triangle right-angled at B. Let P be the point on BC such that BP=PC. If AB = 10 cm, \angle BAP=45^\circ and \angle CAP=\theta<br>(use \tan(\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta})

  1. A. I and II only
  2. B. II and III only
  3. C. I and III only
  4. D. I, II and III ✓

Correct Answer: D. I, II and III

Explanation

I. AP is a median (since BP=PC), so it divides the area of \triangle ABC into two equal parts (True). II. AC = \sqrt{10^2+20^2} \approx 22.36, PC = 10, AP = \sqrt{10^2+10^2} \approx 14.14. Perimeter = 22.36 + 10 + 14.14 = 46.5 \gt 46 (True). III. Area of \triangle APC = \frac{1}{2} \times \text{Area}(\triangle ABC) = \frac{1}{2} \times (\frac{1}{2} \times 10 \times 20) = 50 (True).

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