Question

Douglas has a segment with endpoints I(5, 2) and J(9, 10) that is divided by a point K such that IK and KJ form a 2:3 ratio. He knows that the distance between the x-coordinates is 4 units. Which of the following fractions will let him find the x-coordinate for point K?
2/3, 2/5, 3/2, 3/5

Answer 1

Answer:

The correct option is 2.

Step-by-step explanation:

It is given that the endpoints of the line segment are I(5, 2) and J(9, 10).

According to the section formula, if a point divide the line segment in m:n, then the coordinates of that point are

$(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$

The x-coordinate is also written as

$x-coordinate=\frac{m}{m+n}(x_2-x_1)+x_1$

The distance between the x-coordinates is 4 units and $x_1=5$. So, the  fractions that will let him find the x-coordinate for point K is

$\frac{m}{m+n}=\frac{2}{2+3}=\frac{2}{5}$

The required fraction is 2/5.

The x-coordinate of K is

$x-coordinate=\frac{2}{5}(4)+5=\frac{33}{5}$

Therefore the correct option is 2.

Answer 2

Check the picture below.

$\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ I(5,9)\qquad J(9,10)\qquad \qquad 2:3 \\\\\\ \cfrac{I\underline{K}}{\underline{K} J} = \cfrac{2}{3}\implies \cfrac{I}{J} = \cfrac{2}{3}\implies 3I=2J\implies 3(5,9)=2(9,10) \\\\ -------------------------------$

$\bf { K=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}\\\\ -------------------------------\\\\ K=\left(\cfrac{(3\cdot 5)+(2\cdot 9)}{2+3}\quad ,\quad \cfrac{(3\cdot 9)+(2\cdot 10)}{2+3}\right)$

so. the x-coordinate of K will then be at  $\bf \cfrac{(3\cdot 5)+(2\cdot 9)}{2+3}$

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$\bf \cfrac{(3\cdot 5)+(2\cdot 9)}{2+3}\implies \cfrac{33}{5}\implies 6\frac{3}{5} \\\\\\ I(5,9)\impliedby \textit{to go from 5, to }6\frac{3}{5}\textit{ is just }1\frac{3}{5}\implies \cfrac{8}{5}\implies \cfrac{4\cdot 2}{5} \\\\\\ or\implies 4\cdot \cfrac{2}{5}\impliedby \textit{distance from 5 to }6\frac{3}{5}$