Question

The endpoints of RS are R(−4,3) and S(16,18). Complete each statement using a fraction.
a. (8,12) is the point​ ___ of the way from R to S.
b. (0,6) is the point​ ___ of the way from R to S.

Answer 1

Answer:

a) (8,12) is the point $\frac{3}{5}$ of the way from R to S, b) (0, 6) is the point $\frac{1}{5}$ of the way from R to S.

Step-by-step explanation:

Let be $X$ a point in the line between endpoints $R$ and $S$. The ratio of the line segment $RX$ to the line segment $RS$ is given by:

$r = \frac{RX}{RS}$

Where:

$RX = \sqrt{(x_{X}-x_{R})^{2}+(y_{X}-y_{R})^{2}}$

$RS = \sqrt{(x_{S}-x_{R})^{2}+(y_{S}-y_{R})^{2}}$

Let be $x_{R} = -4$, $y_{R} = 3$, $x_{S} = 16$ and $y_{S} = 18$.

a) If we know that $x_{X} = 8$ and $y_{X} = 12$, the ratio of the line segment $RX$ to the line segment $RS$ is:

$RX =\sqrt{[8-(-4)]^{2}+(12-3)^{2}}$

$RX = 15$

$RS =\sqrt{[16-(-4)]^{2}+(18-3)^{2}}$

$RS = 25$

Then,

$r = \frac{15}{25}$

$r = \frac{3}{5}$

(8,12) is the point $\frac{3}{5}$ of the way from R to S.

b) If we know that $x_{X} = 0$ and $y_{X} = 6$, the ratio of the line segment $RX$ to the line segment $RS$ is:

$RX =\sqrt{[0-(-4)]^{2}+(6-3)^{2}}$

$RX = 5$

$RS =\sqrt{[16-(-4)]^{2}+(18-3)^{2}}$

$RS = 25$

Then,

$r = \frac{5}{25}$

$r = \frac{1}{5}$

(0, 6) is the point $\frac{1}{5}$ of the way from R to S.