Question

ab is parallel to de. find ab in the figure given below.
e 80 cm d
10 cm 20 cm
c
60 cm 30 cm
a b

Explanation:

Step1: Identify Similar Triangles

Since \( AB \parallel DE \), \( \triangle CDE \sim \triangle CAB \) (by AA similarity, as vertical angles at \( C \) are equal and alternate interior angles are equal due to parallel lines).

Step2: Find Ratio of Corresponding Sides

First, find the ratio of sides \( CD \) to \( CA \) and \( CE \) to \( CB \).
\( CD = 20 \, \text{cm} \), \( CA = 60 \, \text{cm} \), so ratio \( \frac{CD}{CA} = \frac{20}{60} = \frac{1}{3} \).
\( CE = 10 \, \text{cm} \), \( CB = 30 \, \text{cm} \), so ratio \( \frac{CE}{CB} = \frac{10}{30} = \frac{1}{3} \).
Thus, the similarity ratio of \( \triangle CDE \) to \( \triangle CAB \) is \( \frac{1}{3} \).

Step3: Use Proportion for \( AB \)

For similar triangles, \( \frac{DE}{AB} = \frac{1}{3} \). Given \( DE = 80 \, \text{cm} \), solve for \( AB \):
\( \frac{80}{AB} = \frac{1}{3} \)
\( AB = 80 \times 3 = 240 \, \text{cm} \).

Answer:

\( 240 \, \text{cm} \)