If the diagonals of a cyclic quadrilateral are equal, then the quadrilateral isa)rhombusb)squarec)rectangled)none of theseCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.

(Consider BD as a chord)
∠BCD + ∠BAD = 180 (Cyclic quadrilateral)

∠BCD = 180− 90 = 90
(Considering AC as a chord)

∠ADC + ∠ABC = 180 (Cyclic quadrilateral)

90+ ∠ABC = 180
∠ABC = 90

Each interior angle of a cyclic quadrilateral is of 90.Hence it is a rectangle.

Answer

Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.

(Consider BD as a chord)
∠BCD + ∠BAD = 180 (Cyclic quadrilateral)

∠BCD = 180− 90 = 90
(Considering AC as a chord)

∠ADC + ∠ABC = 180 (Cyclic quadrilateral)

90+ ∠ABC = 180
∠ABC = 90

Each interior angle of a cyclic quadrilateral is of 90.Hence it is a rectangle.

Answer

See the intersecting point yrr .

Answer

Explanation:

To understand why the correct answer is option 'C' (rectangle), let's first understand what a cyclic quadrilateral is and what properties it possesses.

Cyclic Quadrilateral:
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, meaning that all four vertices of the quadrilateral lie on the circumference of a circle.

Properties of Cyclic Quadrilateral:
1. Opposite angles of a cyclic quadrilateral are supplementary, which means they add up to 180 degrees.
2. The sum of the measures of any two opposite angles of a cyclic quadrilateral is 180 degrees.

Now coming back to the question, if the diagonals of a cyclic quadrilateral are equal, we need to analyze the possible shapes of the quadrilateral.

Possible Shapes:
1. Rhombus: A rhombus is a quadrilateral with all sides equal. In a rhombus, the diagonals are perpendicular bisectors of each other. Since the diagonals of a cyclic quadrilateral are equal, a rhombus can be a possible shape.

2. Square: A square is a special case of a rhombus where all angles are right angles (90 degrees). In a square, the diagonals are equal and bisect each other at right angles. Since the diagonals of a cyclic quadrilateral are equal, a square can be a possible shape.

3. Rectangle: A rectangle is a quadrilateral with all angles equal to 90 degrees. In a rectangle, the diagonals are equal and bisect each other at right angles. Since the diagonals of a cyclic quadrilateral are equal, a rectangle can be a possible shape.

Since both a square and a rectangle satisfy the condition of having equal diagonals in a cyclic quadrilateral, we need to determine which shape is more specific.

Differentiating between Square and Rectangle:
In a square, all sides are equal, whereas in a rectangle, opposite sides are equal. Therefore, if the diagonals of a cyclic quadrilateral are equal, but the sides are not equal, the quadrilateral cannot be a square. However, it can still be a rectangle if the opposite sides are equal.

Therefore, the correct answer is option 'C' (rectangle).

By understanding the properties and characteristics of different shapes, we can determine the correct answer to this question.

Top Courses for Class 10View all

Science Class 10

Social Studies (SST) Class 10

Mathematics (Maths) Class 10

Past Year Papers for Class 10

English Class 10

Top Courses for Class 10

Top Courses for Class 10

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company