Question

In triangle ABC A(5,6) and its centroid is (2,3) then the mid point of the side BC is

Answer 1

The coordinates of mid-point of BC are (1/2,3/2).

Given:

• In ∆ABC, A(5,6).
• It's centroid is (2,3).

To find:

• Find the mid point of the side BC.

Solution:

Concept/Formula to be used:

• Centroid (G) is the intersection of all medians of a triangle.
• Centroid divides the median in 2:1.
• Section formula: If a point (C) divides the line segment joining the end points (x1,y1) and (x2,y2) in m:n, then coordinates of C(x,y) are given by $\bf x = \frac{mx_2 + nx_1}{m + n} \\ \\ \bf y= \frac{my_2 + ny_1}{m + n}$

Step 1:

Write the given values.

Median from vertex A, bisects side BC.

or we can say that, Median is to draw from opposite vertex.

Let the midpoint of BC is P.

It divides AP in 2:1.

Step 2:

Find the coordinates of midpoint of BC.

Coordinates of vertex A(5,6)

Coordinates of centroid G(2,3)

Ratio of internel division is 2:1.

Apply section formula, as P(x,y)

$2 = \frac{2x + 5 \times 1}{2 + 1}$

or

$2x + 5 = 6$

or

$2x = 1$

or

$\bf x = \frac{1}{2}$

by the same way

$3 = \frac{2y +6 }{1 + 2}$

or

$2y + 6 = 9$

or

$2y = 3$

or

$\bf y = \frac{3}{2}$

Thus,

The coordinates of mid-point of BC are (1/2,3/2).

#SPJ3

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Answer 2

Given, in triangle ABC, A(5,6)

Centroid G is (2,3)

We need to find the midpoint of the side BC.

Let's know the meaning of centroid first.

Median: Median is a line which joins the midpoint of a line and the opposite vertex of that line. In the figure AD is a median.

Centroid: Centroid is the point of intersection of three medians.

Centroid divides the median in the ratio 2:1

Here, in this diagram G divides the median AD in the ratio 2:1

Here A is (5,6)

Let D be (x,y)

We know that, the point which divides the line segment joining the points A(x1, y1) and B(x2, y2) in the ratio m:n internally is

$((mx2 + nx1) \div m + n \: \: \: \: \: \: \: (my2 + ny1) \div m + n)$

By substituting the values we get,

$(2 \: \: \: \: 3) = ((2 \times x + 1 \times 5) \div 2 + 1 \: \: \: \: \: \: (2 \times y + 1 \times 6) \div 2 + 1 \\ (2 \: \: \: 3) = ((2x + 5) \div 3 \: \: \: (2y + 6) \div 3)$

Now by equating we get

$2 = (2x + 5 )\div 3 \\ 6 = 2x + 5 \\ 2x = 1 \\ x = 1 \div 2$

Now equating y terms we get

$3 = (2y + 6) \div 3 \\ 9 = 2y + 6 \\ 2y = 3 \\ y = 3 \div 2$

Therefore the midpoint of the side BC is (x,y)= (1/2,3/2)

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