Multiple choice

If a, b and c are the sides of a triangle ABC and equations ax2 + bx + c = 0 and 5x2 + 12x + 13 = 0 have both the roots common, then the given triangle is

  1. acute

  2. obtuse

  3. right

  4. either 1 or 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If ax^2 + bx + c = 0 and 5x^2 + 12x + 13 = 0 have both roots common, then a/5 = b/12 = c/13 = k. Thus a=5k, b=12k, c=13k. Since a, b, c are sides of a triangle, they must satisfy the triangle inequality. However, 5+12=17 > 13, so it is a valid triangle. Checking for a right triangle: 5^2 + 12^2 = 25 + 144 = 169 = 13^2. Since a^2 + b^2 = c^2, it is a right triangle.

AI explanation

For the two quadratic equations ax^2 + bx + c = 0 and 5x^2 + 12x + 13 = 0 to have both roots common, the ratios of their corresponding coefficients must be equal, meaning a/5 = b/12 = c/13 = k. This gives the side lengths of the triangle as a = 5k, b = 12k, and c = 13k. Checking the Pythagorean theorem, (5k)^2 + (12k)^2 = 25k^2 + 144k^2 = 169k^2, which perfectly equals (13k)^2. Because the square of the longest side equals the sum of the squares of the other two sides, the triangle is a right triangle.

Keep practicing — related questions