Q) If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

9th Class Maths – NCERT Important Questions

Ans:

Step 1: Let’s make a diagram for better understanding of the question:

Here AB and PQ are the 2 equal chords and R is the point of intersection.

We need to prove that: BR = PR and AR = QR

Step 2:

Let’s draw perpendicular OC on chord AB and perpendicular OD on chord PQ.

Step 3: Now let’s compare Δ OCR and Δ ODR:

∠OCR = ∠ ODR          (being 90⁰)

OC = OD                     (Equal chords are at equal distance from Center)

OR = OR                     (common side)

∴ Δ OCR Δ ODR

Now by Corresponding Parts of Congruent Triangles (CPCT) rule:

CR = DR ……………..(i)

Step 4: It is given that chords AB = CD

We know that the perpendicular from center on a chord bisects it,

∴ 2 CB = 2 DP

∴ CB = DP ………….. (ii)

and CA = DQ …………..(iii)

Step 5: Now by adding equations (i) and (ii), we get:

CR + CB = DR + DP

BR = PR…………. Hence Proved !

Step 6: Now by subtracting equation (i) from equation (iii), we get:

CA – CR = DQ – DR

AR = QR …………. Hence Proved !

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