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The points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the vertices of ∆ ABC. Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2:1

Solution:

Given, the vertices of ∆ ABC are A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃).

The points Q and R are the points on medians BE and CF such that BQ:QE = 2:1 and

CR:RF = 2:1.

We have to find the coordinates of the points Q and R.

Let the coordinates of Q be (a, b) and R be (p, q).

the-points-a-x1-y1-b-x2-y2-and-c-x3-y3-are-the-vertices-of-abc-find-the-coordinates-of-points-q-and-r-on-medians-be-and-cf-respectively-such-that-bq-qe-2-1

The coordinates of the mid-point of the line segment joining the points P (x₁ , y₁) and Q (x₂ , y₂) are [(x₁ + x₂)/2, (y₁ + y₂)/2]

E is the midpoint of A(x₁, y₁) and C(x₃, y₃)

E = [(x₁ + x₃)/2, (y₁ + y₃)/2]

F is the midpoint of A(x₁, y₁) and B(x₂ , y₂)

F = [(x₁ + x₂)/2, (y₁ + y₂)/2]

By section formula,

The coordinates of the point P which divides the line segment joining the points A (x₁ , y₁) and B (x₂ , y₂) internally in the ratio m₁ : m₂ are

[(m₁x₂ + m₂x₁)/m₁ + m₂ , (m₁y₂ + m₂y₁)/m₁ + m₂]

The point Q divides the line B(x₂ , y₂) and E((x₁ + x₃)/2, (y₁ + y₃)/2) in the ratio 2:1

Coordinates of Q, (a, b) = [2(x₁ + x₃)/2 + 1(x₂)/2 + 1, 2(y₁ + y₃)/2 + 1(y₂)/2 + 1]

(a, b) = [(x₁ + x₂ + x₃)/3, (y₁ + y₂ +y₃)/3]

The point R divides the line C(x₃, y₃) and F((x₁ + x₂)/2, (y₁ + y₂)/2) in the ratio 2:1

Coordinates of R, (p, q) = [2(x₁ + x₃)/2 + 1(x₂)/2 + 1, 2(y₁ + y₃)/2 + 1(y₂)/2 + 1]

(p, q) = [(x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3]

Therefore, the coordinates of Q and R ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) and ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

✦ Try This: A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ABC. D(7/2, 9/2) E(5/2, 3) and F(5, 7/2) are the midpoints of BC, AC and AB respectively. Find the coordinates of points Q and R on medians BE and CF such that BQ : QE = 2 : 1 and CR : RF = 2:1

Given, the vertices of ∆ABC are A (4, 2), B(6, 5) and C(1, 4)

The midpoint of BC, AC and AB are D(7/2, 9/2) E(5/2, 3) and F(5, 7/2).

We have to find the coordinates of points Q and R on medians BE and CF such that BQ:QE = 2:1 and CR:RF = 2:1.

Let the coordinates of Q be (a, b) and R be (p, q).

By section formula,

The coordinates of the point P which divides the line segment joining the points A (x₁ , y₁) and B (x₂ , y₂) internally in the ratio m₁ : m₂ are

[(m₁x₂ + m₂x₁)/m₁ + m₂ , (m₁y₂ + m₂y₁)/m₁ + m₂]

The point Q divides the line B(6, 5) and E(5/2, 3) in the ratio 2:1

Coordinates of Q, (a, b) = [2(5/2) + 1(6)/2 + 1, 2(3) + 1(5)/2 + 1]

(a, b) = [(5 + 6)/3, (6 + 5)/3]

(a, b) = [11/3, 11/3]

The point R divides the line C(1, 4) and F(5, 7/2) in the ratio 2:1

Coordinates of R, (p, q) = [2(5) + 1(1)/2 + 1, 2(7)/2 + 1(4)/2 + 1]

(p, q) = [(10 + 1)/3, (7 + 4)/3]

(p, q) = [11/3, 11/3]

Therefore, the coordinates of Q and R are (11/3, 11/3) and (11/3, 11/3).

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.4 Problem 3(iii)

The points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the vertices of ∆ ABC. Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2:1

Summary:

The points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the vertices of ∆ ABC. The coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF =2:1 are (11/3, 11/3) and (11/3, 11/3)

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