Question

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

Explanation:

Step1: Recall Midsegment Theorem

The Triangle Midsegment Theorem states that a midsegment of a triangle is parallel to the third side and half as long. So, if \( MN \), \( MO \), and \( ON \) are midsegments, then:

• \( MN \) is midsegment, so \( KL = 2 \times MN \). Given \( MN = 2 \), so \( KL = 2\times2 = 4 \)? Wait, no, wait. Wait, let's check the labels. Wait, \( MN \), \( MO \), \( ON \) are midsegments. Wait, \( M \), \( N \), \( O \) are midpoints? Wait, the midsegments: \( MN \) connects midpoints of \( JK \) and \( JL \)? Wait, no, the diagram: \( M \) on \( JK \), \( N \) on \( JL \), \( O \) on \( KL \). So midsegments: \( MN \) is midsegment to \( KL \), \( MO \) is midsegment to \( JL \), \( ON \) is midsegment to \( JK \). Wait, let's re-express:

Midsegment Theorem: Midsegment length = \( \frac{1}{2} \) length of the third side. So:

• If \( MN \) is midsegment (connecting midpoints of \( JK \) and \( JL \)), then \( MN \parallel KL \) and \( MN = \frac{1}{2}KL \). Given \( MN = 2 \), so \( KL = 2 \times 2 = 4 \)? Wait, no, in the diagram, \( MO = 4 \), \( ON = 3.5 \), \( MN = 2 \). Wait, maybe \( M \), \( N \), \( O \) are midpoints, so:

• \( M \) is midpoint of \( JK \), \( N \) midpoint of \( JL \), \( O \) midpoint of \( KL \). Then:

• \( MN \) is midsegment to \( KL \), so \( KL = 2 \times MN \). \( MN = 2 \), so \( KL = 4 \).

• \( MO \) is midsegment to \( JL \), so \( JL = 2 \times MO \). \( MO = 4 \), so \( JL = 8 \).

• \( ON \) is midsegment to \( JK \), so \( JK = 2 \times ON \). \( ON = 3.5 \), so \( JK = 7 \).

Now, perimeter of \( \triangle JKL \) is \( JK + KL + JL \).

So \( JK = 7 \), \( KL = 4 \), \( JL = 8 \)? Wait, no, that can't be. Wait, maybe I mixed up the midsegments. Wait, maybe \( MN \) is midsegment to \( KL \), \( MO \) to \( JL \), \( ON \) to \( JK \). Wait, let's check the lengths:

Wait, the midsegments are \( MN \), \( MO \), \( ON \). So:

• \( MN \) is midsegment: \( MN = \frac{1}{2}KL \) ⇒ \( KL = 2 \times MN = 2 \times 2 = 4 \).

• \( MO \) is midsegment: \( MO = \frac{1}{2}JL \) ⇒ \( JL = 2 \times MO = 2 \times 4 = 8 \).

• \( ON \) is midsegment: \( ON = \frac{1}{2}JK \) ⇒ \( JK = 2 \times ON = 2 \times 3.5 = 7 \).

Now, perimeter of \( \triangle JKL \) is \( JK + KL + JL = 7 + 4 + 8 = 19 \).

Step2: Calculate Perimeter

Sum the three sides: \( JK = 7 \), \( KL = 4 \), \( JL = 8 \). So \( 7 + 4 + 8 = 19 \).

Answer:

19