Question

1. the medians of △def are shown. given eh = 15x - 6 and hf = 11x + 14, find ef.

Explanation:

Step1: Recall median property

In a triangle, a median divides the opposite - side into two equal segments. So, if \(EH\) and \(HF\) are parts of a median of \(\triangle DEF\), then \(EH = HF\).
\[15x−6=11x + 14\]

Step2: Solve for \(x\)

Subtract \(11x\) from both sides:
\[15x-11x−6=11x-11x + 14\]
\[4x−6=14\]
Add 6 to both sides:
\[4x−6 + 6=14 + 6\]
\[4x=20\]
Divide both sides by 4:
\[x=\frac{20}{4}=5\]

Step3: Find \(EH\) or \(HF\)

Substitute \(x = 5\) into the expression for \(EH\) (we could also use the expression for \(HF\)).
\[EH=15x−6=15\times5−6=75 - 6=69\]

Step4: Find \(EF\)

Since \(EF=EH + HF\) and \(EH = HF\), then \(EF = 2EH\) (or \(2HF\)).
\[EF=2\times69 = 138\]

Answer:

138