Sides AB and BC and median AD of a triangle ABC are...

Sides $\mathrm{AB}$ and $\mathrm{BC}$ and median $\mathrm{AD}$ of $\mathrm{a}$
triangle $\mathrm{ABC}$ are respectively proportional to sides $\mathrm{PQ}$ and $\mathrm{QR}$ and median PM of $\Delta$ PQR (see Fig. $6.41$ ). Show that $\Delta \mathrm{ABC}-\Delta \mathrm{PQR}$

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Key Concepts

Median of a Triangle

A median is a line segment drawn from a vertex to the midpoint of the opposite side, thereby dividing the triangle into two regions of equal area. This concept is fundamental in triangle geometry and is often used in proofs regarding triangle properties and congruence.

Proportionality of Segments

Proportionality involves the comparison of ratios of corresponding segments, such as sides or medians, in different geometric figures. When certain segments of one triangle are proportional to corresponding segments of another, this relationship can be used to establish similarity or, under special conditions, congruence.

Triangle Congruence

Triangle congruence is a key concept where two triangles are shown to be identical in shape and size. Congruence criteria, such as the Side-Angle-Side (SAS) postulate, rely on establishing the equality of corresponding parts to conclude that two triangles are congruent.

Scale Factor and Similarity

The scale factor is the constant ratio that exists between the lengths of corresponding sides in similar figures. When triangles have segments in a fixed proportion, they are similar; if the scale factor is 1, the triangles are also congruent, meaning all corresponding sides and angles are equal.

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