Answer

Verified

413.1k+ views

**Hint:**

Here we need to apply the concept of Cyclic Quadrilateral, Parallelogram, Transversal, isosceles triangle.

Cyclic Quadrilateral: A quadrilateral inscribed in a circle in which the sum of opposite angles is ${{180}^{\circ }}$ .

Quadrilateral: Four sided closed figure in which the sum of adjacent angles is ${{180}^{\circ }}$ .

Parallelogram: Four sided closed figure in which opposite sides are parallel and equal.

Isosceles triangle: Two angles are equal in a triangle it becomes isosceles triangle. Sides opposite to two equal angles are equal

Transversal: A line which cuts two or more parallel lines.

\[1=3\text{ },\text{ }5=7,\text{ }2=4\text{ }and\text{ }6=8\] :vertically opposite angles.

Sum of two adjacent angles is ${{180}^{\circ }}$

Sum of two co- interior angles is ${{180}^{\circ }}$i.e\[,\text{ }4+5,6+3\] .

Corresponding angles are equal, \[1=5,\text{ }2=6,3=7\text{ }and\text{ }4=8.\]

We need to find the length of BC.

**Complete step by step solution:**

\[AD\] is transversal to \[AB\] and\[CD\] .

$\Rightarrow \angle A+\angle D={{180}^{\circ }}$ (Sum of two co- interior angles is${{180}^{\circ }}$)…………(1)

\[ABCD\] is a cyclic quadrilateral

\[\Rightarrow \angle A+\angle C={{180}^{\circ }}\]( Sum of two opposite angles is ${{180}^{\circ }}$)……………(2)

From the above equations 1 and 2,

\[\Rightarrow \angle A+\angle D=\angle A+\angle C\]

\[\therefore \angle C=\angle D\] …………………….(3)

Draw a line \[BE\] which is parallel to \[AD\], then it forms a \[\Delta BEC\]

\[DE\] becomes transversal to \[AD\] and \[BE\],

$\Rightarrow \angle BEC=\angle ADE$ ………….(4) (Corresponding angles are equal)

From equation 3 and 4 we can conclude that ,

\[\angle BEC=\angle ADE=\angle BCE\]

\[\therefore \angle BEC=\angle BCE\] ………….(5)

As two angles are equal in a triangle it becomes an isosceles triangle. Sides opposite to two equal angles are equal.

$\Rightarrow BC=BE$ …………….(6)

But \[ABED\] forms a parallelogram (Opposite sides are equal)

\[\Rightarrow =BE=\] ⇒ \[AD\]\[=B\text{ }E=\] \[5\text{ }cm\] ……………….(7)

From equations 6 and 7,

$\Rightarrow $ \[BC=BE=AD=\] \[5\text{ }cm\]

$\therefore $ \[BC\text{ }=\] \[5\text{ }cm\]

Therefore, the length of \[BC\text{ }=\]\[5\text{ }cm\].

Hence, Option choice A is the correct answer.

**Note:**

In such types of questions the concept of Cyclic Quadrilateral, Parallelogram, Transversal, isosceles triangle is needed. Knowledge about the concepts helps in applying the concept to the question. Then it is solved accordingly to get the required value.

Recently Updated Pages

Identify the type of clause underlined in the sentence class 8 english CBSE

Which statement describes the density of the inner class 8 social science CBSE

Babur considered which ruler of Gujarat as among the class 8 social science CBSE

Which island groups were affected by the Tsunami in class 8 social science CBSE

Which is the administrative system that works under class 8 social science CBSE

The year in which the state was named as Karnataka class 8 social science CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail