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 (
) and
(
) in ratio
, then
and
![y = \frac{y_1m_2 + y_2m_1}{m_1 + m_2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7By_1m_2%20%2B%20y_2m_1%7D%7Bm_1%20%2B%20m_2%7D)
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](https://tex.z-dn.net/?f=m_1%20%3A%20m_2)
By using the above ratio formula:
![5 = \frac{-3m_2 + 17m_1}{m_1 + m_2}](https://tex.z-dn.net/?f=5%20%3D%20%5Cfrac%7B-3m_2%20%2B%2017m_1%7D%7Bm_1%20%2B%20m_2%7D)
⇒ ![5m_1 + 5m_2 = -3m_2 + 17m_1](https://tex.z-dn.net/?f=5m_1%20%2B%205m_2%20%3D%20-3m_2%20%2B%2017m_1)
⇒ ![5m_2 + 3m_2 = 17m_1 - 5m_1](https://tex.z-dn.net/?f=5m_2%20%2B%203m_2%20%3D%2017m_1%20-%205m_1)
⇒ ![8m_2 = 12m_1](https://tex.z-dn.net/?f=8m_2%20%3D%2012m_1)
⇒ ![\frac{m_1}{m_2} = \frac{12}{8}](https://tex.z-dn.net/?f=%5Cfrac%7Bm_1%7D%7Bm_2%7D%20%20%3D%20%5Cfrac%7B12%7D%7B8%7D)
⇒ ![\frac{m_1}{m_2} = \frac{3}{2}](https://tex.z-dn.net/?f=%5Cfrac%7Bm_1%7D%7Bm_2%7D%20%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
⇒
= 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](https://tex.z-dn.net/?f=m_1%20%3A%20m_2)
By using the above ratio formula:
![13 = \frac{-3m_2 + 17m_1}{m_1 + m_2}](https://tex.z-dn.net/?f=13%20%3D%20%5Cfrac%7B-3m_2%20%2B%2017m_1%7D%7Bm_1%20%2B%20m_2%7D)
⇒ ![13m_1 + 13m_2 = -3m_2 + 17m_1](https://tex.z-dn.net/?f=13m_1%20%2B%2013m_2%20%3D%20-3m_2%20%2B%2017m_1)
⇒ ![13m_2 + 3m_2 = 17m_1 - 13m_1](https://tex.z-dn.net/?f=13m_2%20%2B%203m_2%20%3D%2017m_1%20-%2013m_1)
⇒ ![16m_2 = 4m_1](https://tex.z-dn.net/?f=16m_2%20%3D%204m_1)
⇒ ![\frac{m_1}{m_2} = \frac{4}{1}](https://tex.z-dn.net/?f=%5Cfrac%7Bm_1%7D%7Bm_2%7D%20%3D%20%5Cfrac%7B4%7D%7B1%7D)
⇒
= 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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