Question

mo, mn, and on are the midsegments of △jkl. what is the perimeter of △jkl? 6 9.5 11.5 19

Explanation:

Step1: Recall Midsegment Theorem

The midsegment of a triangle is parallel to the third side and half as long. So, if \( \overline{MN} \), \( \overline{MO} \), \( \overline{ON} \) are midsegments, then:

• \( MN=\frac{1}{2}KL \), \( MO=\frac{1}{2}JL \), \( ON=\frac{1}{2}JK \)

Step2: Find lengths of \( JK \), \( JL \), \( KL \)

From the diagram:

• \( MN = 2 \), so \( KL = 2\times MN = 2\times2 = 4 \)? Wait, no, wait. Wait, \( ON = 3.5 \)? Wait, no, let's check again. Wait, the midsegments: \( MN \) is a midsegment, so \( MN \parallel KL \) and \( MN=\frac{1}{2}KL \). Wait, in the diagram, \( MN = 2 \), so \( KL = 2\times2 = 4 \)? Wait, no, maybe I mixed up. Wait, \( MO \) is a midsegment: \( MO = 4 \), so \( JL = 2\times MO = 2\times4 = 8 \)? Wait, \( ON = 3.5 \), so \( JK = 2\times ON = 2\times3.5 = 7 \)? Wait, \( MN = 2 \), so \( KL = 2\times MN = 2\times2 = 4 \)? Wait, no, maybe the labels: Let's identify the midsegments. \( M \), \( N \), \( O \) are midpoints. So:
• \( M \) is midpoint of \( JK \), \( N \) midpoint of \( JL \), \( O \) midpoint of \( KL \)? Wait, no, the midsegments are \( MO \), \( MN \), \( ON \). So:
• \( \overline{MN} \): connects midpoints of \( JK \) and \( JL \), so \( MN \parallel KL \) and \( MN=\frac{1}{2}KL \)
• \( \overline{MO} \): connects midpoints of \( JK \) and \( KL \), so \( MO \parallel JL \) and \( MO=\frac{1}{2}JL \)
• \( \overline{ON} \): connects midpoints of \( JL \) and \( KL \), so \( ON \parallel JK \) and \( ON=\frac{1}{2}JK \)

So:

• \( MN = 2 \), so \( KL = 2\times MN = 4 \)
• \( MO = 4 \), so \( JL = 2\times MO = 8 \)
• \( ON = 3.5 \), so \( JK = 2\times ON = 7 \)

Wait, but that can't be. Wait, maybe I got the midsegments wrong. Wait, the diagram: \( MN = 2 \), \( MO = 4 \), \( ON = 3.5 \). So:

• \( MN \) is midsegment, so \( MN = \frac{1}{2}KL \implies KL = 2\times MN = 2\times2 = 4 \)
• \( MO \) is midsegment, so \( MO = \frac{1}{2}JL \implies JL = 2\times MO = 2\times4 = 8 \)
• \( ON \) is midsegment, so \( ON = \frac{1}{2}JK \implies JK = 2\times ON = 2\times3.5 = 7 \)

Now, perimeter of \( \triangle JKL \) is \( JK + JL + KL = 7 + 8 + 4 = 19 \)? Wait, 7 + 8 + 4 = 19. Let's check:

Wait, \( JK = 7 \), \( JL = 8 \), \( KL = 4 \). Then perimeter is \( 7 + 8 + 4 = 19 \). Yes, that matches the option 19.

Step3: Calculate Perimeter

Perimeter \( P = JK + JL + KL \)
We found:

• \( JK = 2\times ON = 2\times3.5 = 7 \)
• \( JL = 2\times MO = 2\times4 = 8 \)
• \( KL = 2\times MN = 2\times2 = 4 \)

So \( P = 7 + 8 + 4 = 19 \)

Answer:

19