Question

Triangle J K L is shown. Angle J K L is a right angle. An altitude is drawn from point K to point M on side L J to form a right angle. The length of K M is 6 and the length of M J is 3. What is the length of line segment KJ?
2 StartRoot 3 EndRoot units 3 StartRoot 2 EndRoot units 3 StartRoot 3 EndRoot units 3 StartRoot 5 EndRoot units

Answer 1

Answer:

$3\sqrt{5}\ units$

Step-by-step explanation:

Theorem 1: The height of right triangle drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of these two segments. Hence,

$KM^2=ML\cdot MJ\\ \\6^2=ML\cdot 3\\ \\3ML=36\\ \\ML=12\ units$

Theorem 2: In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Thus,

$KJ^2=MJ\cdot ML\\ \\KJ^2=3\cdot (3+12)\\ \\KJ^2=3\cdot 15\\ \\KJ^2=45\\ \\KJ=\sqrt{45}=3\sqrt{5}\ units$