Question

9. Use Structure Point K is of the way from
J(4,-5) to L(0, -7).
a. What are the coordinates of K if n = 4?
b. What is a formula for the coordinates of K
for any n?

Answer 1

A line can be divided in segment using ratios. The point at which a line is divided in ratios is calculated using: $(\frac{ax_2+bx_1}{a+b},\frac{ay_2+by_1}{a+b})$.

• When $n = 4$, the coordinates of K is $(3, -5.5)$
• The general formula of K for any n is: $K = [\frac{4n - 4}{n},\frac{-2-5n}{n}]$

Given that:

$J(x_1,y_1) = (4,-5)$

$L(x_2,y_2) = (0,-7)$

The coordinate of K is:

$K = \frac{1}{n}$ from J to L

When $n =4$, we have:

$K_1 = \frac 14$ ---- From J to K

$K_2 = 1 - \frac 14$

$K_2 = \frac 34$ ---- From K to L

Express as ratio

$K_1 : K_2=1/4:3/4$

Cancel out 4

$K_1 : K_2=1:3$

Express properly

$a:b =1:3$

So, the coordinates of K when $n =4$ is calculated using the following line ratio formula.

$K = [\frac{ax_2+bx_1}{a+b},\frac{ay_2+by_1}{a+b}]$

So, we have:

$K = [\frac{1 \times 0+3 \times 4}{1+3},\frac{1 \times -7+3 \times -5}{1+3}]$

$K = [\frac{12}{4},\frac{-22}{4}]$

$K = [3,-5.5]$

Hence, the coordinates of K when $n =4$ is $(3,-5.5)$

The formula for K coordinates for any value of n is as follows:

We have:

$K = [\frac{ax_2+bx_1}{a+b},\frac{ay_2+by_1}{a+b}]$

Where:

$n = a+b$

and

$a = 1$

So, the equation becomes

$K = [\frac{ax_2+bx_1}{n},\frac{ay_2+by_1}{n}]$

Substitute $a = 1$ in $n = a+b$

$n = 1 + b$

Make b the subject

$b =n-1$

Substitute $b =n-1$ and $a = 1$ in $K = [\frac{ax_2+bx_1}{n},\frac{ay_2+by_1}{n}]$

$K = [\frac{1 \times x_2+(n - 1) \times x_1}{n},\frac{1 \times y_2+(n - 1) \times y_1}{n}]$

Substitute $J(x_1,y_1) = (4,-5)$ and $L(x_2,y_2) = (0,-7)$

$K = [\frac{1 \times 0+(n - 1) \times 4}{n},\frac{1 \times -7+(n - 1) \times -5}{n}]$

$K = [\frac{4(n - 1)}{n},\frac{-7-5(n - 1)}{n}]$

$K = [\frac{4n - 4}{n},\frac{-7-5n + 5}{n}]$

Collect like terms

$K = [\frac{4n - 4}{n},\frac{-7 + 5-5n}{n}]$

$K = [\frac{4n - 4}{n},\frac{-2-5n}{n}]$

Hence, the general formula of K for any n is:

$K = [\frac{4n - 4}{n},\frac{-2-5n}{n}]$

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