In triangle $ABC$, point $M$ lies on side $AC$ and point $N$ lies on side $BC$, such that $MN$ is parallel to $AB$ (so triangle $CMN$ is the smaller triangle near vertex $C$, and $ABNM$ is the trapezium formed below it). If the area of trapezium $ABNM$ is twice the area of triangle $CMN$, what is the ratio $CM : AM$?

Question

Inside a triangular park, a flower bed forms a triangle similar to the park's boundary. A uniform path runs around the flower bed, so that each side of the park is exactly double the corresponding side of the flower bed. What is the ratio of the area of the path to the area of the flower bed?

Answer 1

Here is a second way to confirm the ratio, using a direct numeric substitution instead of pure algebraic simplification.

Set the area ratio. Since $MN \parallel AB$, triangles $CMN$ and $CAB$ are similar, so if we call the similarity ratio $r = CM/CA$, then $\text{Area}(CMN)/\text{Area}(CAB) = r^2$. Given trapezium $ABNM = 2 \times \text{Area}(CMN)$, the full triangle $CAB$ has area $3 \times \text{Area}(CMN)$, so $r^2 = 1/3$.
Solve for r numerically. $r = 1/\sqrt3 \approx 0.5774$, meaning $CM$ is about 57.7% of $CA$.
Convert to CM:AM. Taking $CA = 1$ as a convenient unit, $CM \approx 0.5774$ and $AM = CA - CM \approx 0.4226$, so $CM:AM \approx 1.366:1$.
Match to the answer choices. Option C gives $(\sqrt3+1)/2 = (1.732+1)/2 = 1.366$, which matches our numeric ratio exactly.

Both the exact algebraic method and this numeric check agree, so option C is correct. \[ \boxed{\dfrac{\sqrt3+1}{2}} \]

Answer 2

Here is a second way to confirm the ratio, using a direct numeric substitution instead of pure algebraic simplification.

Set the area ratio. Since $MN \parallel AB$, triangles $CMN$ and $CAB$ are similar, so if we call the similarity ratio $r = CM/CA$, then $\text{Area}(CMN)/\text{Area}(CAB) = r^2$. Given trapezium $ABNM = 2 \times \text{Area}(CMN)$, the full triangle $CAB$ has area $3 \times \text{Area}(CMN)$, so $r^2 = 1/3$.
Solve for r numerically. $r = 1/\sqrt3 \approx 0.5774$, meaning $CM$ is about 57.7% of $CA$.
Convert to CM:AM. Taking $CA = 1$ as a convenient unit, $CM \approx 0.5774$ and $AM = CA - CM \approx 0.4226$, so $CM:AM \approx 1.366:1$.
Match to the answer choices. Option C gives $(\sqrt3+1)/2 = (1.732+1)/2 = 1.366$, which matches our numeric ratio exactly.

Both the exact algebraic method and this numeric check agree, so option C is correct. \[ \boxed{\dfrac{\sqrt3+1}{2}} \]

Show Hint

The Correct Option is C

Solution and Explanation

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