CBSE2026Class Class 10 · Mathematics2 Marks · Short✅ Verified
(a) In the given figure, AP, AQ and BC are tangents to the circle with centre O. If AB = 6 cm, AC = 7 cm and BC = 5 cm, then what is the length of AP?
OR
(b) In the given figure, TP is tangent to a circle with centre O. Diameter BA when produced meets the tangent at T. If ∠ABP = 35°, then find the measure of ∠PTA.
✅ Answer & Solution
**(a)** Let the circle touch BC at R, and AB, AC produced at P, Q. Using equal tangent lengths: $BR=BP=x$, $CR=CQ=y$, with $x+y=BC=5$. Also $AP=AB+BP=6+x$ and $AQ=AC+CQ=7+y$, and $AP=AQ$ (tangents from A). So $6+x=7+y \Rightarrow x-y=1$. Solving with $x+y=5$: $x=3,y=2$. $$AP=6+3=9\ cm$$
**(b)** Since BA is a diameter, $\angle APB=90°$ (angle in semicircle). In ΔABP: $\angle ABP=35°$, so $\angle BAP=180-90-35=55°$. By the tangent-chord angle, $\angle TPA=\angle ABP=35°$ (angle in alternate segment). Since T,A,B are collinear, $\angle PAT=180°-\angle BAP=180-55=125°$. In ΔAPT: $$\angle PTA=180-\angle TPA-\angle PAT=180-35-125=20°$$
