Points P(6, 0), Q(2, 8) and R(\(-2\), 4) are vertices of \(\triangle\)PQR. It is given that MN $\parallel$ QR such that \(\frac{PM}{MQ} = \frac{1}{3}\). Using distance formula and ratio formula, show that \(\frac{MN}{QR} = \frac{1}{4}\).
Show Hint
Since \(MN \parallel QR\), the triangles \(\triangle PMN\) and \(\triangle PQR\) are similar.
The ratio of their sides is \(\frac{MN}{QR} = \frac{PM}{PQ}\).
Since \(PM : MQ = 1 : 3\), the total ratio is \(\frac{PM}{PM+MQ} = \frac{1}{1+3} = \frac{1}{4}\).
Using similarity properties is an excellent way to pre-verify your coordinates and calculations!
Step 1: Understanding the Question:
This is a Coordinate Geometry question combined with basic triangle similarity concepts.
We are given the vertices of \(\triangle PQR\) and a line segment \(MN\) parallel to side \(QR\).
We are given the ratio in which \(M\) divides side \(PQ\), which is \(\frac{PM}{MQ} = \frac{1}{3}\).
We need to find the coordinates of points \(M\) and \(N\), calculate the lengths \(MN\) and \(QR\) using the distance formula, and verify their ratio.
Step 2: Key Formula or Approach:
1. Section Formula: The coordinates of a point dividing the line joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m : n\) are:
\[ \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]
2. Since \(MN \parallel QR\), by Thales' theorem, the point \(N\) must also divide the side \(PR\) in the same ratio:
\[ \frac{PN}{NR} = \frac{PM}{MQ} = \frac{1}{3} \]
3. Distance Formula: The distance between any two points is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Detailed Explanation:
1. Find the coordinates of point M:
Point \(M\) divides \(PQ\) in the ratio \(1 : 3\). Here, \(P = (6, 0)\) and \(Q = (2, 8)\).
Using Section Formula with \(m = 1\) and \(n = 3\):
\[ x_M = \frac{1(2) + 3(6)}{1+3} = \frac{2 + 18}{4} = \frac{20}{4} = 5 \]
\[ y_M = \frac{1(8) + 3(0)}{1+3} = \frac{8 + 0}{4} = 2 \]
So, \(M = (5, 2)\).
2. Find the coordinates of point N:
Since \(MN \parallel QR\), \(N\) divides \(PR\) in the same ratio \(1 : 3\). Here, \(P = (6, 0)\) and \(R = (-2, 4)\).
Using Section Formula with \(m = 1\) and \(n = 3\):
\[ x_N = \frac{1(-2) + 3(6)}{1+3} = \frac{-2 + 18}{4} = \frac{16}{4} = 4 \]
\[ y_N = \frac{1(4) + 3(0)}{1+3} = \frac{4 + 0}{4} = 1 \]
So, \(N = (4, 1)\).
3. Calculate the length of MN using Distance Formula:
\[ MN = \sqrt{(4 - 5)^2 + (1 - 2)^2} \]
\[ MN = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
4. Calculate the length of QR using Distance Formula:
Here, \(Q = (2, 8)\) and \(R = (-2, 4)\):
\[ QR = \sqrt{(-2 - 2)^2 + (4 - 8)^2} \]
\[ QR = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
5. Find the ratio of MN to QR:
\[ \frac{MN}{QR} = \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{4} \]
Step 4: Final Answer:
Using section and distance formulas, the coordinates of \(M\) and \(N\) were found to be \((5, 2)\) and \((4, 1)\) respectively, which gives \(MN = \sqrt{2}\) and \(QR = 4\sqrt{2}\).
Thus, the ratio \(\frac{MN}{QR}\) is shown to be \(\frac{1}{4}\).
