Question

In the diagram below of triangle JKL, M is the midpoint of JL and N is the midpoint of KL. If MN = -5x + 25, and JK = 35 + 5x,

what is the measure of MN?

Answer 1

`\sf MN = 20`

Explanation:

In triangle JKL, where M is the midpoint of JL and N is the midpoint of KL, the midsegment MN is defined as the line segment that connects the midpoints of two sides. Additionally, the midsegment is half the length of the third side.

Given that
`\sf MN = -5x + 25` and
`\sf JK = 35 + 5x`, we can set up the equation based on the definition of midsegments:

`\sf MN = (1)/(2) JK`

Substitute the given values:

`\sf -5x + 25 = (1)/(2)(35 + 5x)`

Now, solve for x:

Multiply both sides by 2 to eliminate the fraction:

`\sf -10x + 50 = 35 + 5x`

Combine like terms:

`\sf -10x - 5x = 35 - 50`

`\sf -15x = -15`

Divide by -15 to solve for x:

`\sf ( -15x )/(-15)=( -15)/(-15)`

`\sf x = 1`

Now that we know the value of x, substitute it back into the expression for MN:

`\sf MN = -5x + 25`

`\sf MN = -5(1) + 25`

`\sf MN = 20`

So, according to the definition of midsegments, the measure of MN in triangle JKL is
`\sf 20`.