Question

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consider \\( \triangle a b c \\) where \\( d \\) is the midpoint of \\( \overline{a b} \\) and \\( f \\) is the midpoint of \\( \overline{a c} \\). in order to start the proof of the triangle midsegment theorem, which triangles should jamal prove are similar?
option #1: \\( \triangle a d f \\) and \\( \triangle d b c \\)
option #2: \\( \triangle f d c \\) and \\( \triangle d b c \\)
option #3: \\( \triangle f d c \\) and \\( \triangle a b c \\)
option #4: \\( \triangle a d f \\) and \\( \triangle a b c \\)
(1 point)
jamal should prove that the triangles described in option # are similar

Explanation:

Step1: Recall the definition of mid - point

Since \(D\) is the mid - point of \(AB\), then \(\frac{AD}{AB}=\frac{1}{2}\). Since \(F\) is the mid - point of \(AC\), then \(\frac{AF}{AC}=\frac{1}{2}\).

Step2: Use the Side - Angle - Side (SAS) similarity criterion

In \(\triangle ADF\) and \(\triangle ABC\), \(\frac{AD}{AB}=\frac{AF}{AC}=\frac{1}{2}\) and \(\angle A=\angle A\) (common angle). By the SAS similarity criterion (\(\frac{a}{a'}=\frac{b}{b'}\) and the included angles are equal), \(\triangle ADF\sim\triangle ABC\).

Answer:

Option #4. \(\triangle ADF\) and \(\triangle ABC\)