Question

triangle opq is formed by connecting the midpoints of the side of triangle lmn. the lengths of the sides of triangle opq are shown. find the perimeter of triangle lmn. figures not necessarily drawn to scale.

Explanation:

Step1: Recall Midsegment Theorem

The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So, each side of triangle \( LMN \) is twice the length of the corresponding side of triangle \( OPQ \).

Step2: Identify sides of \( \triangle OPQ \)

From the diagram, the sides of \( \triangle OPQ \) are \( 3 \), \( 3 \), and \( 4 \).

Step3: Find sides of \( \triangle LMN \)

• For the side corresponding to \( 3 \) in \( \triangle OPQ \): Length in \( \triangle LMN \) is \( 2\times3 = 6 \).
• For the side corresponding to \( 4 \) in \( \triangle OPQ \): Length in \( \triangle LMN \) is \( 2\times4 = 8 \).

Step4: Calculate perimeter of \( \triangle LMN \)

The sides of \( \triangle LMN \) are \( 6 \), \( 6 \), and \( 8 \) (since two sides correspond to the length \( 3 \) in \( \triangle OPQ \)). Perimeter \( P = 6 + 6 + 8 = 20 \)? Wait, no, wait. Wait, let's re - check. Wait, triangle \( OPQ \) has sides \( 3 \), \( 3 \), \( 4 \). So the sides of \( LMN \): each midsegment is half the length of the third side. So if \( PQ = 3 \), then the side of \( LMN \) parallel to \( PQ \) is \( 2\times3 = 6 \). If \( QO = 3 \), then the side of \( LMN \) parallel to \( QO \) is \( 2\times3 = 6 \). If \( OP = 4 \), then the side of \( LMN \) parallel to \( OP \) is \( 2\times4 = 8 \). Wait, no, maybe I misidentified the sides. Wait, actually, triangle \( OPQ \) is formed by midpoints, so \( O \), \( P \), \( Q \) are midpoints. So \( PQ \) is midsegment of \( \triangle LMN \), so \( LM = 2\times PQ \), \( MN = 2\times QO \), \( NL = 2\times OP \). Wait, let's label the triangle properly. Let's say \( P \) is midpoint of \( MN \), \( Q \) is midpoint of \( NL \), \( O \) is midpoint of \( LM \). Then \( PQ \parallel LM \) and \( PQ=\frac{1}{2}LM \), \( QO \parallel MN \) and \( QO = \frac{1}{2}MN \), \( OP \parallel NL \) and \( OP=\frac{1}{2}NL \). So sides of \( \triangle OPQ \): \( PQ = 3 \), \( QO = 3 \), \( OP = 4 \). Then:

• \( LM = 2\times PQ=2\times3 = 6 \)
• \( MN = 2\times QO = 2\times3=6 \)
• \( NL = 2\times OP = 2\times4 = 8 \)

Perimeter of \( \triangle LMN=LM + MN+NL=6 + 6+8 = 20 \)? Wait, no, wait, maybe I made a mistake. Wait, let's check again. Wait, the sides of \( \triangle OPQ \) are \( 3 \), \( 3 \), \( 4 \). So the perimeter of \( \triangle OPQ \) is \( 3 + 3+4=10 \). Since \( \triangle OPQ \) is the medial triangle (formed by midpoints), its perimeter is half of the perimeter of \( \triangle LMN \). So perimeter of \( \triangle LMN = 2\times(3 + 3+4)=2\times10 = 20 \). Yes, that's a better way. The medial triangle's perimeter is half the perimeter of the original triangle. So if perimeter of \( \triangle OPQ \) is \( 3 + 3+4 = 10 \), then perimeter of \( \triangle LMN \) is \( 2\times10=20 \).

Answer:

\( 20 \)