# A(3, 1), B(Y, 4) and C(1, X) Are Vertices of a Triangle Abc. P, Q and R Are Mid - Points of Sides Bc, Ca and Ab Respectively. Show that the Centroid Of δPqr is the Same as the Centroid δAbc. - Mathematics

Sum

A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid - points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.

#### Solution

Centroid of ΔABC = ((3 + y + 1)/3, (1+ 4 + x)/3) = ((4 + y)/3, (5 + x)/3)

P, Q and R are the mid points of the sides BC, CA and AB.

By mid - point formula, we get

=> P = ((y + 1)/2, (4 + x)/2), Q = (4/2, (1 + x)/2) and R = ((3 + y)/2, 5/2)

Centroid of a ΔPQR = (((y + 1)/2 + 4/2 + (3 + y)/2)/3, ((4 + x)/2 - (1 + x)/2 + 5/2)/3)

= (((y + 1 + 4 + 3 + y)/2)/3, ((4 + x + 1 + x + 5)/2)/3)

= ((8 + 2y)/6, (10 + 2x)/6)

= ((4 + y)/3, (5 + x)/3)               ........(ii)

From (i) and (ii), we get

Centroid of a ∆ABC = Centroid of a ∆PQR

Concept: The Mid-point of a Line Segment (Mid-point Formula)
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#### APPEARS IN

Selina Concise Maths Class 10 ICSE
Chapter 13 Section and Mid-Point Formula
Exercise 13 (C) | Q 22 | Page 183