Question

What are the coordinates of the point 3/5 of the way from A(-9,3) to B(21, -2)

Answer 1

Answer:  The required co-ordinates of the point are (9, 0).

Step-by-step explanation:  We are given to find the co-ordinates of the point that is $\dfrac{3}{5}$ of the way from A(-9,3) to B(21, -2).

Let K be the required point. Then, we mus have

$AK:AB=3:5\\\\\Rightarrow \dfrac{AK}{AK+BK}=\dfrac{3}{5}\\\\\\\Rightarrow 5AK=3AK+3BK\\\\\Rightarrow 2AK=3BK\\\\\Rightarrow AK:BK=3:2.$

We know that

the co-ordinates of a point that divides the line joining the points (a, b) and (c, d) in the ratio m : n are given by

$\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{2}\right).$

For the given division, m : n = 3 : 2.

Therefore, the co-ordinates of the point K are

$\left(\dfrac{3\times21+2\times(-9)}{3+2},\dfrac{3\times(-2)+2\times3}{3+2}\right)\\\\\\=\left(\dfrac{63-18}{5},\dfrac{-6+6}{5}\right)\\\\=\left(\dfrac{45}{5},\dfrac{0}{5}\right)\\\\=(9,0).$

Thus, the required co-ordinates of the point are (9, 0).

Answer 2

Building and solving an equation, it is found that the coordinates are: (9,0).

• We are given two points: A(-9,3) and B(21, -2).
• We also want point C(x,y).

C is 3/5 of the way from A to B, thus:

$C - A = \frac{3}{5}(B - A)$

This is used to find both the x-coordinate and the y-coordinate of C.

First, the x-coordinate, considering $C = x, A = -9, B = 21$.

$C - A = \frac{3}{5}(B - A)$

$x + 9 = \frac{3}{5}(21 + 9)$

$x = 18 - 9$

$x = 9$

Then, the y-coordinate, considering $C = y, A = 3, B = -2$.

$C - A = \frac{3}{5}(B - A)$

$y - 3 = \frac{3}{5}(-2 - 3)$

$y = -3 + 3$

$y = 0$

Thus, the coordinates are (9,0).

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