Question:
ABCD is a parallelogram such that AF = 7 cm, FB = 3 cm and EF = 4 cm, length FD = equals
ABCD is a parallelogram such that AF = 7 cm, FB = 3 cm and EF = 4 cm, length FD = equals
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Whenever you see a line intersecting two parallel lines (such as the opposite sides of a parallelogram) and forming an "X" shape at an intersection point, the two triangles meeting at that point are always similar.
Setting up the ratio of the corresponding opposite sides of the "X" directly gives you the answer!
Setting up the ratio of the corresponding opposite sides of the "X" directly gives you the answer!
Updated On: Jul 9, 2026
- \(\frac{21}{4}\) cm
- \(\frac{28}{3}\) cm
- \(\frac{12}{7}\) cm
- 5.5 cm
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The topic of this question is Similar Triangles.
We are given a parallelogram \(ABCD\) with diagonals and intersecting lines.
The line segments are \(AF = 7 \text{ cm}\), \(FB = 3 \text{ cm}\), and \(EF = 4 \text{ cm}\).
We need to determine the length of the segment \(FD\) by using the properties of similar triangles.
Step 2: Key Formula or Approach:
In a parallelogram \(ABCD\), the opposite sides are parallel. Therefore, we have:
\[ AD \parallel BC \] Since \(E\) lies on the line segment \(BC\), the line segment \(AD\) is parallel to \(BE\).
This parallelism allows us to identify alternate interior angles and establish similarity between \(\Delta ADF\) and \(\Delta EBF\) using the AA similarity criterion.
Once the similarity is proven, the ratio of their corresponding sides will be equal:
\[ \frac{FD}{FB} = \frac{AF}{EF} \]
Step 3: Detailed Explanation:
• Establish the geometric relationships:
Since \(ABCD\) is a parallelogram, the opposite side \(AD\) is parallel to \(BC\).
Since \(E\) is a point on \(BC\), it follows that \(AD \parallel BE\).
• Compare \(\Delta ADF\) and \(\Delta EBF\):
- \(\angle AFD = \angle EFB\) (Vertically opposite angles)
- \(\angle DAF = \angle FEB\) (Alternate interior angles, since \(AD \parallel BE\) and \(AE\) acts as a transversal line)
- \(\angle ADF = \angle FBE\) (Alternate interior angles, since \(AD \parallel BE\) and \(BD\) acts as a transversal line)
• Apply the AA similarity criterion:
Since the corresponding angles are equal, the triangles are similar:
\[ \Delta ADF \sim \Delta EBF \]
• Write down the ratio of their corresponding sides:
\[ \frac{FD}{FB} = \frac{AF}{EF} \]
• Substitute the given segment lengths (\(AF = 7 \text{ cm}\), \(FB = 3 \text{ cm}\), \(EF = 4 \text{ cm}\)) into the equation:
\[ \frac{FD}{3} = \frac{7}{4} \]
• Solve for \(FD\):
\[ FD = \frac{7 \times 3}{4} \] \[ FD = \frac{21}{4} \text{ cm} = 5.25 \text{ cm} \]
Step 4: Final Answer:
The length of \(FD\) is \(\frac{21}{4}\) cm.
Therefore, the correct option is (A).
The topic of this question is Similar Triangles.
We are given a parallelogram \(ABCD\) with diagonals and intersecting lines.
The line segments are \(AF = 7 \text{ cm}\), \(FB = 3 \text{ cm}\), and \(EF = 4 \text{ cm}\).
We need to determine the length of the segment \(FD\) by using the properties of similar triangles.
Step 2: Key Formula or Approach:
In a parallelogram \(ABCD\), the opposite sides are parallel. Therefore, we have:
\[ AD \parallel BC \] Since \(E\) lies on the line segment \(BC\), the line segment \(AD\) is parallel to \(BE\).
This parallelism allows us to identify alternate interior angles and establish similarity between \(\Delta ADF\) and \(\Delta EBF\) using the AA similarity criterion.
Once the similarity is proven, the ratio of their corresponding sides will be equal:
\[ \frac{FD}{FB} = \frac{AF}{EF} \]
Step 3: Detailed Explanation:
• Establish the geometric relationships:
Since \(ABCD\) is a parallelogram, the opposite side \(AD\) is parallel to \(BC\).
Since \(E\) is a point on \(BC\), it follows that \(AD \parallel BE\).
• Compare \(\Delta ADF\) and \(\Delta EBF\):
- \(\angle AFD = \angle EFB\) (Vertically opposite angles)
- \(\angle DAF = \angle FEB\) (Alternate interior angles, since \(AD \parallel BE\) and \(AE\) acts as a transversal line)
- \(\angle ADF = \angle FBE\) (Alternate interior angles, since \(AD \parallel BE\) and \(BD\) acts as a transversal line)
• Apply the AA similarity criterion:
Since the corresponding angles are equal, the triangles are similar:
\[ \Delta ADF \sim \Delta EBF \]
• Write down the ratio of their corresponding sides:
\[ \frac{FD}{FB} = \frac{AF}{EF} \]
• Substitute the given segment lengths (\(AF = 7 \text{ cm}\), \(FB = 3 \text{ cm}\), \(EF = 4 \text{ cm}\)) into the equation:
\[ \frac{FD}{3} = \frac{7}{4} \]
• Solve for \(FD\):
\[ FD = \frac{7 \times 3}{4} \] \[ FD = \frac{21}{4} \text{ cm} = 5.25 \text{ cm} \]
Step 4: Final Answer:
The length of \(FD\) is \(\frac{21}{4}\) cm.
Therefore, the correct option is (A).
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