Question

In a circle with centre O, chord SR = chord SM. Radius OS intersects the chord RM at P. Prove that RP = PM​

Answer 1

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Answer 2

RP = PM​

• A chord of a circle is a section of a straight line whose ends both fall on an arc of a circle. A secant line, often known as secant, is a chord's infinite line extension. A chord is, more broadly speaking, a line segment connecting two points on any curve, such as an ellipse.
• The diameter of a circle is the length of a chord through its centre. The word chord is derived from the Latin word chorda, which means bowstring.

We first need to join RO and OM.

Now, according to the given information, we are given that,

In a circle with centre O, chord SR = chord SM.

Then angles ROS and SOM are equal as they are the equal angles subtended by the equal chords at centre O.

Now, in triangles ROP and MOP, we get that,

OR and OM being radiuses are equal.

OP is a common element to both triangles.

Then, by side-angle-side axiom of congruency, both triangles are congruent.

Hence, RP = PM​ as they are equal parts of congruent triangles.

Hence, proved.

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