Question

A circle is inscribed in a right triangle ABC, right angled at C. The circle is tangent to the segment AB at D and length of segments AD and DB are 7 and 13 respectively. Area of triangle ABC is equal to

• A. 91
• B. 96
• C. 100
• D. 104

Answer 1

Answer: (A)

91

Answer 2

The problem involves a circle inscribed in a right triangle. We need to find the area of the triangle using the given lengths of segments on the hypotenuse.

1. Understand the Geometry and Properties: Let the right triangle be ABC, with the right angle at C.
Let the inscribed circle (incircle) be tangent to the sides BC, AC, and AB at points F, E, and D respectively.
The lengths of tangents from an external point to a circle are equal.

• From vertex A: AD = AE = 7 (given AD = 7).
• From vertex B: BD = BF = 13 (given DB = 13).
• From vertex C: Since C is the right angle, and CE and CF are tangents from C, and IE and IF are radii perpendicular to the sides, the quadrilateral CEIF is a square. Therefore, CE = CF = r, where r is the inradius.

2. Express Side Lengths in Terms of 'r': Let the sides opposite to vertices A, B, C be a, b, c respectively.

• Hypotenuse c = AB = AD + DB = 7 + 13 = 20.
• Leg AC = b = AE + EC = 7 + r.
• Leg BC = a = BF + FC = 13 + r.

3. Apply the Pythagorean Theorem: For a right triangle, $a^2 + b^2 = c^2$.
Substitute the expressions for a, b, and c: $(13 + r)^2 + (7 + r)^2 = 20^2$

4. Solve for 'r': Expand the terms: $(169 + 26r + r^2) + (49 + 14r + r^2) = 400$
Combine like terms: $2r^2 + 40r + 218 = 400$
Subtract 400 from both sides: $2r^2 + 40r - 182 = 0$
Divide the entire equation by 2: $r^2 + 20r - 91 = 0$

5. Calculate the Area of the Triangle: The area of a triangle (Area) can be calculated using the inradius (r) and the semi-perimeter (s) by the formula: Area = rs.

• Semi-perimeter s = (a + b + c) / 2 $s = \frac{(13 + r) + (7 + r) + 20}{2}$ $s = \frac{40 + 2r}{2}$ $s = 20 + r$

• Now, substitute s into the area formula: Area = $r \times (20 + r)$ Area = $r^2 + 20r$

• From the quadratic equation we derived: $r^2 + 20r - 91 = 0$.
  This implies $r^2 + 20r = 91$.

• Therefore, the Area of triangle ABC = 91.