Question

If triangle PQR is formed by joining the mid points of the sides of triangle ABC, then the ratio of the areas of triangle PQR and triangle ABC is

Answer 1

Answer:

The ratio of the area of the triangles is 1:4

Answer 2

The ratio is 1:4

Step-by-step explanation:

The line joining the midpoints of the sides of the triangle form four triangles, each of which is similar to the original triangle.

ΔABC ~ ΔPQR

In ΔABC, P and R are mid points of AB and AC respectively.

∴ PR || BC (midpoint theorem)

In ΔABC and ΔAPR

∠A is common and ∠APR = ∠ABC (corresponding angles)

Therefore, ΔABC ~ ΔAPR (AA similarity)

In ΔABC and ΔPQR,

since P, Q, R are the midpoints of AB, BC and AC respectively,

using  (midpoint theorem)

$PR =\frac{1}{2} BC$

∴ ΔABC ~ ΔPQR (SSS similarity)

$\frac{ar(\bigtriangleup PQR )}{ar(\bigtriangleup ABC )}= \frac{PR^2}{BC^2}= \frac{1}{4}$

Hence , The ratio is 1:4

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