Question

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 \(\frac{MN}{QR} = \frac{1}{4}\).

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!

Answer 1

Correct Answer: 4

Step 1: Find coordinates of M using the Section Formula.
M divides PQ in ratio PM:MQ = 1:3. Using the section formula with P(6,0) and Q(2,8): \(M = \left(\frac{1 \times 2 + 3 \times 6}{1+3}, \frac{1 \times 8 + 3 \times 0}{1+3}\right) = \left(\frac{20}{4}, \frac{8}{4}\right) = (5, 2)\).
Step 2: Find coordinates of N using the Section Formula.
Since MN \(\parallel\) QR, by BPT, N divides PR in ratio PN:NR = 1:3. With P(6,0) and R(-2,4): \(N = \left(\frac{1 \times (-2) + 3 \times 6}{4}, \frac{1 \times 4 + 3 \times 0}{4}\right) = \left(\frac{16}{4}, \frac{4}{4}\right) = (4, 1)\).
Step 3: Calculate MN using distance formula.
\(MN = \sqrt{(5-4)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2}\).
Step 4: Calculate QR using distance formula.
\(QR = \sqrt{(2-(-2))^2 + (8-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\).
Step 5: Find the ratio MN/QR.
\(\frac{MN}{QR} = \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{4}\).
Step 6: Conclusion - the required ratio is proved.
\[ \boxed{\dfrac{MN}{QR} = \dfrac{1}{4}} \]