Question

ABCD is a trapezium in which AB || DC and E, F are points on AD and BC respectively such that EF || DC. If ED = 36 cm, BF = 70 cm and FC = 30 cm, then the length of AD is :

• A. 124 cm
• B. 120 cm
• C. 110 cm
• D. 114 cm

Hint:

Always read carefully whether the question asks for the missing segment (\(AE\)) or the entire side (\(AD\)).
Here, \(AE = 84 \text{ cm}\) is not the final answer; we must add \(36 \text{ cm}\) to get \(120 \text{ cm}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a trapezium \(ABCD\) with \(AB \parallel DC\). Points \(E\) and \(F\) are on the non-parallel sides \(AD\) and \(BC\) respectively such that \(EF \parallel DC\). We are given the lengths of three segments and need to find the total length of the side \(AD\).

Step 2: Key Formula or Approach:
By the Intercept Theorem (or Thales's basic proportionality theorem applied to trapeziums), any line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally:
\[ \frac{AE}{ED} = \frac{BF}{FC} \]

Step 3: Detailed Explanation:

• Identify the given values:
\(ED = 36 \text{ cm}\)
\(BF = 70 \text{ cm}\)
\(FC = 30 \text{ cm}\)

• Substitute these values into the proportionality relation:
\[ \frac{AE}{36} = \frac{70}{30} \]

• Simplify the fraction on the right-hand side:
\[ \frac{AE}{36} = \frac{7}{3} \]

• Solve for \(AE\):
\[ AE = \frac{7}{3} \times 36 = 7 \times 12 = 84 \text{ cm} \]

• Find the total length of \(AD\) by adding \(AE\) and \(ED\):
\[ AD = AE + ED \]
\[ AD = 84 + 36 = 120 \text{ cm} \]


Step 4: Final Answer:
The length of \(AD\) is \(120 \text{ cm}\).