In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If ∠APB=60∘, then the length of PA is :
Get the complete, step-by-step math solution for: "In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If APB = 60^ , then the length of PA is :". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify properties of tangents and angles
When a tangent is drawn to a circle, the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, angles ∠OAP and ∠OBP are both 90∘.
Step 2: Find angle AOP using quadrilateral properties
Consider the quadrilateral OAPB. The sum of angles in a quadrilateral is 360∘. Since ∠OAP=90∘, ∠OBP=90∘, and ∠APB=60∘, we can find ∠AOB=360∘−(90∘+90∘+60∘)=360∘−240∘=120∘. Alternatively, the angle between the two tangents from an external point and the angle subtended by the chord joining the points of contact at the center are supplementary, so ∠AOB=180∘−60∘=120∘.
Step 3: Determine angle APO
The line segment PO bisects ∠APB and ∠AOB. Therefore, ∠APO is half of ∠APB.
Step 4: Use trigonometry in right-angled triangle OAP
In the right-angled triangle △OAP, we know the angle ∠APO=30∘ and the side OA=5 cm (radius). We can use the tangent function, which is the ratio of the opposite side to the adjacent side.
Step 5: Calculate the length of PA
Substitute the value of tan(30∘)=31 into the equation and solve for PA.