Points \(P(6, 0)\), \(Q(2, 8)\) and \(R(-2, 4)\) are vertices of \(\Delta PQR\). 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
Alternatively, you can prove this using similar triangles.
Since \(MN \parallel QR\), we have \(\Delta PMN \sim \Delta PQR\) by AA similarity.
Therefore, the ratio of any two corresponding sides is equal to the ratio of any other two corresponding sides:
\[ \frac{MN}{QR} = \frac{PM}{PQ} \]
Since \(\frac{PM}{MQ} = \frac{1}{3}\), we have \(\frac{PM}{PQ} = \frac{1}{1 + 3} = \frac{1}{4}\).
Thus, \(\frac{MN}{QR} = \frac{1}{4}\) directly without finding coordinates or using the distance formula!
Step 1: Understanding the Question:
We are given a triangle \(\Delta PQR\) with vertices \(P(6, 0)\), \(Q(2, 8)\), and \(R(-2, 4)\).
A line segment \(MN\) is drawn parallel to \(QR\), with \(M\) on \(PQ\) and \(N\) on \(PR\).
The point \(M\) divides the segment \(PQ\) in the ratio \(1 : 3\).
We need to find the coordinates of \(M\) and \(N\) using the section formula, and then use the distance formula to show that the ratio of the lengths \(\frac{MN}{QR}\) is equal to \(\frac{1}{4}\).
Step 2: Key Formula or Approach:
1. Basic Proportionality Theorem (BPT):
Since \(MN \parallel QR\), the line segment \(MN\) divides \(PR\) in the same ratio as \(PQ\).
\[ \frac{PN}{NR} = \frac{PM}{MQ} = \frac{1}{3} \]
2. Section Formula:
The coordinates of a point dividing the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) internally in the ratio \(m_1 : m_2\) are:
\[ \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right) \]
3. Distance Formula:
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Detailed Explanation:
• Let us find the coordinates of point \(M\) on segment \(PQ\).
- Point \(M\) divides \(PQ\) internally in the ratio \(1 : 3\).
- Given \(P(6, 0)\) and \(Q(2, 8)\), with \(m_1 = 1\) and \(m_2 = 3\):
\[ x_M = \frac{1(2) + 3(6)}{1 + 3} = \frac{2 + 18}{4} = \frac{20}{4} = 5 \]
\[ y_M = \frac{1(8) + 3(0)}{1 + 3} = \frac{8 + 0}{4} = 2 \]
- So, the coordinates of \(M\) are \((5, 2)\).
• Let us find the coordinates of point \(N\) on segment \(PR\).
- Since \(MN \parallel QR\), by the Basic Proportionality Theorem, \(N\) also divides \(PR\) internally in the ratio \(1 : 3\).
- Given \(P(6, 0)\) and \(R(-2, 4)\), with \(m_1 = 1\) and \(m_2 = 3\):
\[ x_N = \frac{1(-2) + 3(6)}{1 + 3} = \frac{-2 + 18}{4} = \frac{16}{4} = 4 \]
\[ y_N = \frac{1(4) + 3(0)}{1 + 3} = \frac{4 + 0}{4} = 1 \]
- So, the coordinates of \(N\) are \((4, 1)\).
• Calculate the length of \(MN\) using the distance formula:
- Points are \(M(5, 2)\) and \(N(4, 1)\):
\[ MN = \sqrt{(4 - 5)^2 + (1 - 2)^2} \]
\[ MN = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
• Calculate the length of \(QR\) using the distance formula:
- Points are \(Q(2, 8)\) and \(R(-2, 4)\):
\[ QR = \sqrt{(-2 - 2)^2 + (4 - 8)^2} \]
\[ QR = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
• Calculate the ratio of \(\frac{MN}{QR}\):
\[ \frac{MN}{QR} = \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{4} \]
- Hence, the ratio is successfully shown to be \(\frac{1}{4}\).
Step 4: Final Answer:
The ratio of \(MN\) to \(QR\) is proved to be \(\frac{1}{4}\).
