If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
Solution:
Consider a cyclic quadrilateral ABCD
A pair of opposite sides of a cyclic quadrilateral are equal.
Given, AD = BC
Considering triangle AOD and BOC,
We know that the same segment subtends equal angle to the circle
∠OAD = ∠OBC
Given, AD = BC
∠ODA = ∠OCB
ASA criterion states that two triangles are congruent, if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.
By ASA criterion, the triangles AOD and BOC are similar.
△AOD ⩬ △BOC
Adding triangle DOC on both sides,
△AOD + △DOC ⩬ △BOC + △DOC
△ADC ⩬ △BCD
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are similar, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
AC = BD
Therefore, it is proven that if a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal.
✦ Try This: Prove that a diameter of a circle which bisect a chord of the circle also bisect the angle subtended by the chord at the centre of the circle.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.3 Problem 12
If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
Summary:
If a pair of opposite sides of a cyclic quadrilateral are equal, it is proven that its diagonals are also equal
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