In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If $\angle NML =…

Question

In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If $\angle NML = 30^\circ$, then $\angle RLM$ is :

Accepted Answer

## Model Answer **Option (B) 60°** Since MN is a diameter, ∠MLN = 90° (angle in a semicircle). In △MNL, ∠NLM = 90° − 30° = 60°. Since RS is tangent at L, OL ⊥ RS, so ∠RLM = 90° − ∠NLM = 90° − 60° = **60°**. Source: Chapter 10, Tangent to a Circle --- ## Explanation - **Angle in a semicircle:** ∠MLN = 90° (angle subtended by diameter at any point on circle). - In △NML: ∠NLM = 180° − 90° − 30° = 60°. - **Tangent ⊥ radius (Theorem 10.1):** The radius OL ⊥ RS, so ∠OLR = 90°. Since OL lies along the diameter direction (N–M), ∠RLM = 90° − ∠NLM = 30°... **Correction (careful reasoning):** OL is the radius to L, perpendicular to RS. The angle ∠RLN + ∠NLM + ... Note that ∠RLM = 90° − ∠NLM only if N is on the R-side. From the figure, ∠RLM = 90° − ∠(between OL and LM). Since ∠NML = 30°, arc reasoning gives ∠RLM = ∠NML = **60°** by the tangent-chord angle theorem (alternate segment theorem): the angle between tangent and chord equals the inscribed angle in the alternate segment. **Key rule to rem…

Mathematics — Standard (041) — CBSE Class 10 board question