Question:
In the given figure, $AB \parallel EF$. If $AB = 24\text{ cm}$, $EF = 36\text{ cm}$ and $DA = 7\text{ cm}$, then $AE$ equals
In the given figure, $AB \parallel EF$. If $AB = 24\text{ cm}$, $EF = 36\text{ cm}$ and $DA = 7\text{ cm}$, then $AE$ equals
Show Hint
Always simplify the ratio of parallel sides ($\frac{24}{36} = \frac{2}{3}$) first before performing algebraic cross-multiplication.
This keeps the numerical calculations simple and error-free!
This keeps the numerical calculations simple and error-free!
Updated On: Jul 22, 2026
- $2.5\text{ cm}$
- $10.5\text{ cm}$
- $3.5\text{ cm}$
- $\frac{14}{3}\text{ cm}$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We are given two parallel line segments $AB \parallel EF$.
The lines intersect such that we have two nested triangles $\Delta DAB$ and $\Delta DEF$.
The lengths are $AB = 24\text{ cm}$, $EF = 36\text{ cm}$, and $DA = 7\text{ cm}$.
We need to find the length of the segment $AE$.
Step 2: Key Formula or Approach:
Since $AB \parallel EF$, the corresponding angles are equal:
- $\angle DAB = \angle DEF$ (Corresponding angles)
- $\angle DBA = \angle DFE$ (Corresponding angles)
Also, the angle $\angle D$ is common to both triangles.
Therefore, by AA similarity:
\[ \Delta DAB \sim \Delta DEF \]
Thus, the corresponding sides are proportional:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]
Step 3: Detailed Explanation:
• Express the length $DE$ as a sum of its collinear parts:
Let $AE = x$. Then:
\[ DE = DA + AE = 7 + x \]
• Set up the similarity proportion:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]
Substitute the given values ($DA = 7$, $AB = 24$, $EF = 36$):
\[ \frac{7}{7 + x} = \frac{24}{36} \]
• Simplify the right-hand fraction:
\[ \frac{24}{36} = \frac{2}{3} \]
So:
\[ \frac{7}{7 + x} = \frac{2}{3} \]
• Solve the equation for $x$ by cross-multiplying:
\[ 7 \times 3 = 2(7 + x) \]
\[ 21 = 14 + 2x \]
\[ 2x = 21 - 14 \]
\[ 2x = 7 \implies x = 3.5\text{ cm} \]
Step 4: Final Answer:
The length of $AE$ is $3.5\text{ cm}$.
We are given two parallel line segments $AB \parallel EF$.
The lines intersect such that we have two nested triangles $\Delta DAB$ and $\Delta DEF$.
The lengths are $AB = 24\text{ cm}$, $EF = 36\text{ cm}$, and $DA = 7\text{ cm}$.
We need to find the length of the segment $AE$.
Step 2: Key Formula or Approach:
Since $AB \parallel EF$, the corresponding angles are equal:
- $\angle DAB = \angle DEF$ (Corresponding angles)
- $\angle DBA = \angle DFE$ (Corresponding angles)
Also, the angle $\angle D$ is common to both triangles.
Therefore, by AA similarity:
\[ \Delta DAB \sim \Delta DEF \]
Thus, the corresponding sides are proportional:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]
Step 3: Detailed Explanation:
• Express the length $DE$ as a sum of its collinear parts:
Let $AE = x$. Then:
\[ DE = DA + AE = 7 + x \]
• Set up the similarity proportion:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]
Substitute the given values ($DA = 7$, $AB = 24$, $EF = 36$):
\[ \frac{7}{7 + x} = \frac{24}{36} \]
• Simplify the right-hand fraction:
\[ \frac{24}{36} = \frac{2}{3} \]
So:
\[ \frac{7}{7 + x} = \frac{2}{3} \]
• Solve the equation for $x$ by cross-multiplying:
\[ 7 \times 3 = 2(7 + x) \]
\[ 21 = 14 + 2x \]
\[ 2x = 21 - 14 \]
\[ 2x = 7 \implies x = 3.5\text{ cm} \]
Step 4: Final Answer:
The length of $AE$ is $3.5\text{ cm}$.
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