M32-06

Official Solution:


If triangle ABC is right-angled at vertex A, what is the area of triangle ABC?

The area of a right triangle is \(\frac{leg_1* leg_2}{2} = \frac{AB *AC}{2}\). Essentially, to answer the question, we need to determine the value of AB*AC.

(1) AB + AC = 8.

This statement is clearly insufficient. There can be an infinite number of combinations for the lengths.

(2) The length of the longest side of the triangle is \(5\sqrt{2}\).

The longest side of a right triangle is the hypotenuse. However, knowing the length of the hypotenuse alone isn't enough to determine the area. This statement is also not sufficient.

(1)+(2) Square (1): \((AB)^2 + 2(AB)(AC) + (AC)^2 = 8^2\). Using the Pythagorean theorem, we can deduce that \((AB)^2 +(AC)^2 = (BC)^2 = (5 \sqrt{2})^2 = 50\). Therefore, \((AB)^2 + 2(AB)(AC) + (AC)^2 = 50 + 2(AB)(AC) = 8^2\). From this, we find that \((AB)(AC) = 7\). Sufficient.


Answer: C