Q) Find the coordinates of the centroid P of the Δ ABC, whose vertices are A(-1, 3), B(3, -1) and C(0, 0). Hence, find the equation of a line passing through P and parallel to AB.

ICSE Specimen Question Paper (SQP)2025

Ans:

(a) Coordinates of the Centroid:

Given that two vertices of a triangle are A(-1, 3), B(3, -1) and C(0, 0).

Let the centroid coordinates be (X, Y)

We know that if a triangle has 3 coordinates as (x₁, y₁), (x₂, y₂), (x₃, y₃);

then the centroid is given by:

∴ for coordinates of Centroid =

=

Therefore, the coordinates of the centroid are:

(b) Equation of the line parallel to AB:

Step 1: Let’s first calculate gradient or slope of the line AB

We know that the slope of a line passing through points (x₁, y₁) and (x₂, y₂) is given by:

m =

∴ slope of line AB, m_AB passing through points  A(-1, 3), B(3, -1) will be:

m_AB =

=

= -1

Step 2: Now the line parallel to AB will have same slop i.e. its m = – 1

(note: if slope of 2nd line is not same as 1st line, it will intersect the 1st line; which should not be the case of parallel lines)

Hence, the equation of the line parallel to AB and passing through point (x₁, y₁) is given by:

y – y₁ = m (x – x₁)

Step 3: Since it is given that this line is passing through the centroid and centroid coordinates are

∴ y – = (- 1) ( x – )

∴ y – = –  x +

∴ y + x =

∴ 3 x + 3 y = 4

Therefore, the equation of the line is 3 x + 3 y = 4. 

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