Question

In the given figure, CM and RN are respectively the medians of $\Delta ABC$ and $\Delta PQR$. If $\Delta ABC \sim \Delta PQR$, then prove that : (i) $\Delta AMC \sim \Delta PNR$, (ii) $\Delta CMB \sim \Delta RNQ$.

Hint:

When working with medians in similar triangles, remember that the ratio of corresponding medians is the same as the ratio of corresponding sides. In this case, $CM/RN = AB/PQ = AC/PR = BC/QR$.

Answer 1

Step 1: Understanding the Concept:
When two triangles are similar, their corresponding angles are equal and their corresponding sides are in the same ratio. Medians divide the side they are drawn to into two equal parts.
Step 2: Key Formula or Approach:
SAS (Side-Angle-Side) Similarity Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.
Step 3: Detailed Explanation:
Part (i): Prove $\Delta AMC \sim \Delta PNR$
Given $\Delta ABC \sim \Delta PQR$.
Therefore, $\angle A = \angle P$ and $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} \dots(1)$.
Since $CM$ is the median of $\Delta ABC$, $M$ is the mid-point of $AB$, so $AB = 2AM$.
Since $RN$ is the median of $\Delta PQR$, $N$ is the mid-point of $PQ$, so $PQ = 2PN$.
Substitute these into the ratio from (1):
\[ \frac{2AM}{2PN} = \frac{AC}{PR} \implies \frac{AM}{PN} = \frac{AC}{PR} \]
In $\Delta AMC$ and $\Delta PNR$:
1. $\frac{AM}{PN} = \frac{AC}{PR}$ (Proved above)
2. $\angle A = \angle P$ (Given)
By SAS similarity criterion, $\Delta AMC \sim \Delta PNR$.
Part (ii): Prove $\Delta CMB \sim \Delta RNQ$
Similarly, $AB = 2MB$ and $PQ = 2NQ$.
From (1), $\frac{2MB}{2NQ} = \frac{BC}{QR} \implies \frac{MB}{NQ} = \frac{BC}{QR}$.
In $\Delta CMB$ and $\Delta RNQ$:
1. $\frac{MB}{NQ} = \frac{BC}{QR}$ (Proved above)
2. $\angle B = \angle Q$ (Corresponding angles of similar $\Delta ABC$ and $\Delta PQR$)
By SAS similarity criterion, $\Delta CMB \sim \Delta RNQ$.
Step 4: Final Answer:
Both (i) and (ii) are proved using the SAS similarity criterion and properties of medians.