If **ABCD** is a trapezium in which **AB** is parallel to **CD**, the sides **AD** and **BC** when extended intersect at point **E**, and given that **AB = 2 cm**, **CD = 1 cm**, and the perimeter of **ABCD** is **6 cm**, then the perimeter of **△AEB**, in cm, is

Question

Accepted Answer

We are tasked with finding the perimeter of $$ 	riangle AEB $$ given the following information:

- $$ ABCD $$ is a trapezium with $$ AB \parallel CD $$.
- $$ AB = 2 \, 	ext{cm} $$, $$ CD = 1 \, 	ext{cm} $$, and the perimeter of $$ ABCD = 6 \, 	ext{cm} $$.
- The sides $$ AD $$ and $$ BC $$ are extended to meet at point $$ E $$.

#### Step 1: Assign variables to unknown sides
Let:
- $$ AD = x $$,
- $$ BC = 3 - x $$ (since the total perimeter is $$ AB + CD + AD + BC = 6 $$).

#### Step 2: Identify similar triangles
The problem states that $$ AB \parallel CD $$, so triangles $$ 	riangle AEB $$ and $$ 	riangle DEC $$ are similar. Additionally:
- The length of $$ AB = 2x $$, since $$ D $$ is the midpoint of $$ AE $$,
- The length of $$ BE = 6 - 2x $$, since $$ C $$ is the midpoint of $$ EB $$.

#### Step 3: Calculate the lengths of sides in $$ 	riangle AEB $$
The sides of $$ 	riangle AEB $$ are:
- $$ AB = 2 \, 	ext{cm} $$,
- $$ AE = 2x, $$
- $$ BE = 6 - 2x. $$

#### Step 4: Compute the perimeter of $$ 	riangle AEB $$
The perimeter of $$ 	riangle AEB $$ is:
$$
	ext{Perimeter} = AB + AE + BE
$$
Substitute the values:
$$
	ext{Perimeter} = 2 + 2x + (6 - 2x)
$$
Simplify:
$$
	ext{Perimeter} = 2 + 6 = 8.
$$

### Final Answer:
**Option C:** **8 cm**

Answer

Answer

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