If MN, NO, and MO are midsegments, find the perimeter of JKL.
Understand the Problem
The question is asking to find the perimeter of a geometric figure (specifically a triangle) using the lengths of its midsegments. This involves understanding the relationships between the sides of the triangle and applying the midsegment theorem.
Answer
The perimeter of triangle $JKL$ is $540$.
Answer for screen readers
The perimeter of triangle ( JKL ) is ( 540 ).
Steps to Solve
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Identify the Midsegments The midsegments ( MN ), ( NO ), and ( MO ) are segments connecting the midpoints of the sides of triangle ( JKL ). According to the midsegment theorem, each midsegment is half the length of the side it is parallel to.
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Set Up the Equations From the diagram, we have:
- ( MN ) is parallel to side ( JK )
- ( NO ) is parallel to side ( KL )
- ( MO ) is parallel to side ( JL )
Based on the diagram:
- ( MN = 84 ) (half of ( JK ))
- ( NO = 98 ) (half of ( KL ))
- ( MO = 88 ) (half of ( JL ))
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Express the Sides in Terms of the Midsegments Using the midsegment theorem:
- ( JK = 2 \times MN = 2 \times 84 = 168 )
- ( KL = 2 \times NO = 2 \times 98 = 196 )
- ( JL = 2 \times MO = 2 \times 88 = 176 )
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Calculate the Perimeter Now, we find the perimeter of triangle ( JKL ) by adding the lengths of its sides: [ \text{Perimeter} = JK + KL + JL = 168 + 196 + 176 ]
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Perform the Addition Compute the total perimeter: [ 168 + 196 + 176 = 540 ]
The perimeter of triangle ( JKL ) is ( 540 ).
More Information
The perimeter is the total distance around a geometric figure. In the case of triangles, this involves adding the lengths of all three sides. The midsegment theorem is an important property of triangles that helps in finding side lengths when only midsegment lengths are given.
Tips
- Forgetting to multiply the midsegment length by 2 to find the corresponding side length.
- Misapplying the midsegment theorem, thinking that midsegments are the actual lengths of the triangle sides rather than half of them.
- Errors in addition; ensure careful addition of the three lengths to find the correct perimeter.
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