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In the given figure, $X Y ‖ B C$ . Find the length of $X Y, given B C = 6$ $cm$ .

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6 cm

4 cm

1.5 cm

1 cm

answer is C.

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Detailed Solution

Given that,

X Y ‖ B C

.
The side

X Y

Δ A X Y

is parallel to side

B C

Δ A B C

As per the theorem of corresponding angles, the angles formed due to the intersection of transversal on the parallel lines

X Y

and

B C

are congruent to each other.
Transversal

A B

intersects the side

X Y

and

B C

.
The angles formed are

∠ A X Y

and

∠ A B C

.
Hence,

∠ A X Y = ∠ A B C

…… (1)
Transversal

A C

intersects the side

X Y

and

B C

.
The angles formed are

∠ A Y X

and

∠ A C B

.
Hence,

∠ A Y X = ∠ A C B

…… (2)
Consider

Δ A X Y

and

Δ A B C

,
From equation (1) and (2), by AA similarity test,

Δ A B C ~ Δ A X Y

A B B C = A X X Y

……(3)
Side

X Y

has intersected the side

A B

in two parts.
AB = AX + BX
As per the information given in the question, AX = 1cm

a n d B X = 3

cm.
Substitute the value of AX and BX in the equation to find the value of AB.

A B = 1 + 3 = 4 = 4 cm

…(4)
From the equation (3), we get,

A B B C = A X X Y

As per the information given in the question,

A X = 1

cm,

B C = 6

cm.
From equation (4),

A B = 4

cm.
Substitute the

A X = 1

cm,

B C = 6

, a n d A B = 4

cm in equation (3).

⇒ 46 = 1 X Y ⇒ X Y = 64 ⇒ X Y = 1.5 cm

The value of XY is 1.5 cm.
Hence, option 3) is correct.

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