In parallelogram MNOP, side MN is extended to Q such that MN = NQ. Line PQ intersects ON at R. Find the ratio in which R divides ON.

Question

Accepted Answer

Step 1: Let us assign coordinates to the vertices of parallelogram MNOP for ease of calculation.

Assume M at origin (0,0).
Let vector MN = vector a, so N is at (a,0).
Let vector MP = vector b, so P is at (0,b).
Since MNOP is a parallelogram, O is at N + MP = (a,b).

Step 2: Given MN = NQ, and Q lies on the line extended from MN beyond N.
Since MN = a units along x-axis, N is at (a,0), so Q is at (2a,0).

Step 3: Find equation of line PQ.
P is at (0,b), Q is at (2a,0).
Slope of PQ = (0 - b)/(2a - 0) = -b/(2a).
Equation of PQ: y - b = (-b/(2a))(x - 0) => y = b - (b/(2a))x.

Step 4: Find equation of ON.
O is at (a,b), N is at (a,0).
ON is vertical line x = a.

Step 5: Find intersection R of PQ and ON.
At x = a, y = b - (b/(2a)) * a = b - b/2 = b/2.
So R is at (a, b/2).

Step 6: Find ratio in which R divides ON.
ON runs from O (a,b) to N (a,0).
Length ON = b.
R divides ON into OR and RN.
OR = distance from O to R = b - b/2 = b/2.
RN = distance from R to N = b/2.

Ratio OR : RN = (b/2) : (b/2) = 1 : 1.

But the options do not have 1:1 as correct answer. Let's re-check the vector assignment.

Alternate approach:
Let M = (0,0), N = (1,0) (unit vector along x-axis).
Let MP = vector b = (x,y).
Then O = N + MP = (1 + x, y).

Q is such that MN = NQ, so Q is at (2,0).

Line PQ passes through P (x,y) and Q (2,0).
Equation of PQ:
Slope = (0 - y)/(2 - x) = -y/(2 - x)
Equation: y - y = (-y/(2 - x))(x - x) => y = y - (y/(2 - x))(x - x) (trivial)

We want to find intersection R of PQ with ON.
ON is line from O (1 + x, y) to N (1,0).
Parametric form of ON:
X = 1 + x - t*x
Y = y - t*y
For t=0, point is O; for t=1, point is N.

At intersection R, coordinates satisfy PQ and ON.
From PQ equation:
(y - y) = (-y/(2 - x))(x - x) => trivial.

Alternatively, parametrize PQ:
PQ vector = Q - P = (2 - x, 0 - y) = (2 - x, -y)
Parametric form:
X = x + s(2 - x)
Y = y + s(-y) = y - s y

At intersection R, coordinates equal on both lines:
X = 1 + x - t x = x + s(2 - x)
Y = y - t y = y - s y

From Y:
y - t y = y - s y => t y = s y => t = s

From X:
1 + x - t x = x + t (2 - x)
1 + x - t x = x + t (2 - x)
1 + x - t x = x + 2 t - t x
1 + x - t x - x + t x = 2 t
1 = 2 t
So t = 1/2

Since t = s, s = 1/2

Therefore, R divides ON at t = 1/2, meaning OR : RN = t : (1 - t) = 1/2 : 1/2 = 1 : 1.

Hence, the ratio is 1 : 1.

Since option 4 is 1 : 1, it is the correct answer.

Final answer: 1 : 1.

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