← See all Geometry problems

Find the length of a side of a right triangle given one leg and two angles.

Check the final answer first, then review the worked steps.

Problem

Find the length of a side of a right triangle given one leg and two angles.

Answer

\(9\sqrt{2}\)

Step-by-step solution

  1. Identify the triangle type: The image shows a triangle with angles measuring $45^\circ$ and $45^\circ$. Since the sum of angles in a triangle is $180^\circ$, the third angle is $180^\circ - 45^\circ - 45^\circ = 90^\circ$. This confirms it is a right triangle. Furthermore, because two angles are equal ($45^\circ$), it is an isosceles right triangle.
  2. Understand the properties of an isosceles right triangle: In an isosceles right triangle, the two legs (the sides opposite the acute angles) are equal in length. The hypotenuse (the side opposite the right angle) is $\sqrt{2}$ times the length of a leg.
  3. Relate the given information to the triangle: The image shows one leg of the triangle is $9\sqrt{2}$ km. The side labeled 'm' is the other leg. Since it's an isosceles right triangle, the two legs must be equal.
  4. Determine the length of side m: Therefore, the length of side $m$ is equal to the length of the other leg, which is $9\sqrt{2}$ km.
  5. State the final answer: The value of $m$ is $9\sqrt{2}$ km.