(a) If a hexagon PQRSTU circumscribes a circle, prove that, PQ + RS + TU = QR + ST + UP.
OR
(b) In the given figure, two concentric circles have radii 3 cm and 5 cm. Two tangents TR and TP are drawn to the circles from an external point T such that TR touches the inner circle at R and TP touches the outer circle at P. If TR = 410 cm, then find the length of TP.
Get the complete, step-by-step math solution for: "(a) If a hexagon PQRSTU circumscribes a circle, prove that, PQ + RS + TU = QR + ST + UP. OR (b) In the given figure, two concentric circles have radii...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify given information and properties of tangents
We are given the radii of the inner and outer concentric circles as OR=3 cm and OP=5 cm respectively. We are also given the length of the tangent TR=410 cm. A key property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, ∠ORT=90∘ and ∠OPT=90∘.
Step 2: Calculate the length of OT using the Pythagorean theorem in △ ORT
In the right-angled triangle △ORT, we can use the Pythagorean theorem to find the length of OT. Substituting the given values, we have OT2=(3)2+(410)2.
Step 3: Solve for OT
Calculating the squares, 32=9 and (410)2=16×10=160. Adding these gives OT2=9+160=169. Taking the square root, we find OT=13 cm.
Step 4: Calculate the length of TP using the Pythagorean theorem in △ OPT
Now consider the right-angled triangle △OPT. We know OT=13 cm and OP=5 cm. Using the Pythagorean theorem, we can find the length of the tangent TP.
Step 5: Solve for TP
Substituting the values, TP2=132−52=169−25=144. Taking the square root, we get TP=12 cm.