Question

The endpoints of RS are R(-3,3) and S(17,13) . Complete each statement using a fraction.
A: 5,7 is the point ___ of the way from R to S
B: 13,11 is the point ___ of the way from R to S

Answer 1

A: 5,7 is the point 3 : 2 of the way from R to S

B: 13,11 is the point 4 : 1 of the way from R to S

Let point (x, y) divides the line segment whose end points are ($x_1, y_1$) and

($x_2, y_2$) in ratio $m_1 : m_2$, then

$x = \frac{x_1m_2 + x_2m_1}{m_1 + m_2}$ and

$y = \frac{y_1m_2 + y_2m_1}{m_1 + m_2}$

Solving for A:

According to the question, RS is the line segment with end points R (-3, 3) and S (17, 13) and point (5, 7) divides in the ratio $m_1 : m_2$

By using the above ratio formula:

$5 = \frac{-3m_2 + 17m_1}{m_1 + m_2}$

⇒ $5m_1 + 5m_2 = -3m_2 + 17m_1$

⇒ $5m_2 + 3m_2 = 17m_1 - 5m_1$

⇒ $8m_2 = 12m_1$

⇒ $\frac{m_1}{m_2} = \frac{12}{8}$

⇒ $\frac{m_1}{m_2} = \frac{3}{2}$

⇒ $m_1 : m_2$ = 3 : 2

Solving for B:

According to the question, RS is the line segment with end points R (-3, 3) and S (17, 13) and point (13, 11) divides in the ratio $m_1 : m_2$

By using the above ratio formula:

$13 = \frac{-3m_2 + 17m_1}{m_1 + m_2}$

⇒ $13m_1 + 13m_2 = -3m_2 + 17m_1$

⇒ $13m_2 + 3m_2 = 17m_1 - 13m_1$

⇒ $16m_2 = 4m_1$

⇒ $\frac{m_1}{m_2} = \frac{4}{1}$

⇒ $m_1 : m_2$ = 4 : 1

∴ Point (5, 7) divides the line segment RS in the ratio 3 : 2.

∴ Point (13, 11) divides the line segment RS in the ratio 4 : 1.

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