Question

please solve the above question​

Answer 1

$\large\underline{\text{Theorems or Important Facts}}$

$\red{\bigstar}$Midpoint theorem.

The three points of a triangle we are going to choose are two midpoints. After we join them, the ratio of two corresponding segments is $2:1$.

$\red{\bigstar}$Similarity.

The triangle formed by connecting midpoints is similar to the original triangle, because all corresponding sides have a ratio of $2:1$, and by SSS similarity postulate.

$\large\underline{\text{Solution}}$

By midpoint theorem, the similarity ratio of the new triangle to the original triangle is $2:1$.

As similarity ratio is $2:1$, the ratio of the shape is $2^{2}:1^{2}=4:1$.

So option (c) is correct.

Answer 2

Answer: (C) 4 : 1.

Explaination:

Here, ∆ABC ~ ∆DEF

(As, AB/DE = BC/EF = AC/DF = 2:1 by mid-point theorem.)

And we know,

Ratio of area of ∆s = (Ratio of sides corresponding to the triangles)²

=> ratio = (2/1)² = 4/1 or 4 : 1 (c)

More:

In two similar triangles,

Ratio of sides = ratio of altitudes = ratio of medians = ratio of perimeter = √(ratio of area).