Question

1. if f, g, and h are the midpoints of the sides of δjkl, fg = 37, kl = 48, and gh = 30, find each measure.

a) fh = ______
b) jl = ______
c) kj = ______
d) fj = ______

Explanation:

Step1: Use the Mid - segment Theorem

The Mid - segment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.

Part (a)

Since \(FG\) is a mid - segment parallel to \(JH\) and \(GH\) is a mid - segment parallel to \(FK\), and \(FH\) is a mid - segment parallel to \(KL\).
By the Mid - segment Theorem, \(FH=\frac{1}{2}KL\).
Given \(KL = 48\), then \(FH=\frac{1}{2}\times48\)
\(FH = 24\)

Part (b)

Since \(FG\) is a mid - segment, and \(FG=\frac{1}{2}JL\) (by the Mid - segment Theorem).
Given \(FG = 37\), then \(JL=2\times FG\)
\(JL=2\times37\)
\(JL = 74\)

Part (c)

Since \(GH\) is a mid - segment, and \(GH=\frac{1}{2}KJ\) (by the Mid - segment Theorem).
Given \(GH = 30\), then \(KJ=2\times GH\)
\(KJ=2\times30\)
\(KJ = 60\)

Part (d)

Since \(F\) is the mid - point of \(KJ\) (because \(F\) is a mid - point of a side of \(\triangle JKL\) and using the Mid - segment properties), and \(KJ = 60\) (from part (c)).
Then \(FJ=\frac{1}{2}KJ\)
\(FJ=\frac{1}{2}\times60\)
\(FJ = 30\)

Answer:

a) \(FH = 24\)
b) \(JL = 74\)
c) \(KJ = 60\)
d) \(FJ = 30\)