Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity

Chapter 3 – CIrcle- Text Book Solution

Practice Set 3.1 | Q 3 | Page 55

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity

solution

Proof: In ∆RMO and ∆RNO,

∠RMO ≅ ∠RNO = 90°   ……[Tangent theorem]

hypt OR ≅ hypt OR    ……[Common side]

seg OM ≅ seg ON    ……[Radii of the same circle]

∴ ∆RMO ≅ ∆RNO      ……[By Hypotenuse side test]

∠MOR ≅ ∠NOR

Similarly ∠MRO ≅ ∠NRO    ……[Corresponding angles of congruent triangles]

Explanation:-

n right triangles RMO and RNO,

• ∠RMO is congruent to ∠RNO and both are 90 degrees (by the Tangent theorem)
• OR is congruent to OR (common side)
• OM is congruent to ON (radii of the same circle) Therefore, by the Hypotenuse-Leg congruence theorem, triangles RMO and RNO are congruent. As a result, ∠MOR is congruent to ∠NOR (corresponding angles of congruent triangles). Similarly, ∠MRO is congruent to ∠NRO.

Chapter 3 – Circle – Text Book Solution

Practice set 3.1  | Q 3 | Page 55

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