Question

right triangle similarity
what is the length of line segment kj?
3√3 units
3√2 units
2√3 units
3√5 units

Explanation:

Step1: Identify the geometric theorem

We use the geometric mean (or altitude-on-hypotenuse) theorem in right triangles. In a right triangle, when an altitude is drawn to the hypotenuse, the length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse segment adjacent to that leg and the entire hypotenuse. But more directly, for right triangle \( \triangle KJM \) (right - angled at \( M \)) and \( \triangle KJL \) (right - angled at \( K \)), we can use the Pythagorean theorem in \( \triangle KJM \). Wait, actually, the altitude - on - hypotenuse theorem states that \( KM^{2}=LM\times MJ \), but we want to find \( KJ \). In right triangle \( \triangle KJM \), we know \( KM = 6 \) and \( MJ=3 \), and we can use the Pythagorean theorem which states that for a right triangle with legs \( a \) and \( b \) and hypotenuse \( c \), \( c^{2}=a^{2}+b^{2} \). Here, in \( \triangle KJM \), \( \angle M = 90^{\circ} \), so \( KJ^{2}=KM^{2}+MJ^{2} \)

Step2: Substitute the values

Given \( KM = 6 \) and \( MJ = 3 \), substitute into the Pythagorean theorem formula: \( KJ^{2}=6^{2}+3^{2}=36 + 9=45 \)

Step3: Solve for \( KJ \)

Take the square root of both sides: \( KJ=\sqrt{45}=\sqrt{9\times5}=3\sqrt{5} \) units

Answer:

\( 3\sqrt{5} \) units