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