Question

1. if f, g, and h are the midpoints of the sides of \\( \triangle j k l, f g = 37, k l = 48 \\), and \\( g h = 30 \\), find each measure.

a) \\( f h = \\)
b) \\( j l = \\)
c) \\( k j = \\)
d) \\( f j = \\)

Explanation:

Step1: Use the Midsegment Theorem

The Midsegment Theorem states that the midsegment of a triangle is parallel to the third side and half its length.

Step2: Find \(FH\)

Since \(FG\) is a midsegment parallel to \(KL\), \(FH\) is a midsegment parallel to \(JL\). But wait, no, actually, \(GH\) is a midsegment parallel to \(KJ\). And \(FG\) is a midsegment parallel to \(KL\). Wait, no, correction: In \(\triangle JKL\), \(F, G, H\) are mid - points. The Midsegment Theorem: If \(M\) and \(N\) are mid - points of two sides of a triangle, then \(MN=\frac{1}{2}\times\) the length of the third side.
For part (a):
\(FH\) is a midsegment. \(FH=\frac{1}{2}KL\). Given \(KL = 48\), so \(FH=\frac{1}{2}\times48=24\)

Step3: Find \(JL\)

\(FG\) is a midsegment. By the Midsegment Theorem, \(FG=\frac{1}{2}JL\). Given \(FG = 37\), then \(JL = 2\times FG\). So \(JL=2\times37 = 74\)

Step4: Find \(KJ\)

\(GH\) is a midsegment. By the Midsegment Theorem, \(GH=\frac{1}{2}KJ\). Given \(GH = 30\), then \(KJ=2\times GH\). So \(KJ = 2\times30=60\)

Step5: Find \(FJ\)

Since \(F\) is the mid - point of \(KJ\) (by the definition of a midsegment, as \(F\) and \(H\) are mid - points of two sides of \(\triangle JKL\)), and \(KJ = 60\), then \(FJ=\frac{1}{2}KJ\). So \(FJ=\frac{1}{2}\times60 = 30\)

Answer:

a) \(24\)
b) \(74\)
c) \(60\)
d) \(30\)