Question

what is the length of line segment kj?
options:
3√3 units
3√2 units
2√3 units
3√5 units

Explanation:

Step1: Identify the right triangle

We have right triangle \( \triangle KMJ \) with \( \angle KMJ = 90^\circ \), \( KM = 6 \), and \( MJ = 3 \). We need to find the length of \( KJ \).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( c = \sqrt{a^2 + b^2} \). Here, \( a = KM = 6 \) and \( b = MJ = 3 \)? Wait, no, wait. Wait, actually, \( KM = 6 \), \( MJ = 3 \), and \( KJ \) is the hypotenuse? Wait, no, looking at the diagram, \( KM \) is 6, \( MJ \) is 3, and \( \angle KMJ \) is right angle. Wait, no, maybe I misread. Wait, the right angle is at \( M \), so \( KM \) and \( MJ \) are the legs? Wait, no, \( KM = 6 \), \( MJ = 3 \), so \( KJ \) is the hypotenuse. Wait, no, wait, let's check again. Wait, the triangle is \( KMJ \), right-angled at \( M \), so \( KM = 6 \), \( MJ = 3 \), so \( KJ = \sqrt{KM^2 + MJ^2} \)? Wait, no, that would be \( \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \)? Wait, but that's one of the options. Wait, let's confirm.

Wait, the diagram shows \( KM = 6 \), \( MJ = 3 \), and \( \angle KMJ = 90^\circ \). So by Pythagoras, \( KJ^2 = KM^2 + MJ^2 \). So \( KJ = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \).

Answer:

\( 3\sqrt{5} \) units (corresponding to the option with \( 3\sqrt{5} \) units)