Tangents and Secants to a Circle — Telangana (SSC) Class 10 Mathematics Solutions (Free)
Free step-by-step Telangana (SSC) Class 10 Mathematics solutions for "Tangents and Secants to a Circle" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Telangana (SSC) Class 10 Mathematics solutions for "Tangents and Secants to a Circle" — important questions with detailed answers, d…
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Q1: From an external point P, a tangent PT is drawn to circle with center O and radius 5 cm. If OP = 13 cm, find the length of tangent PT.
Step 1: A tangent is perpendicular to the radius at the point of contact
Step 2: Triangle OTP is a right triangle with right angle at T
Step 3: OT = radius = 5 cm, OP = 13 cm
Step 4: Use Pythagoras: OP² = OT² + PT²
Step 5: 13² = 5² + PT²
Step 6: 169 = 25 + PT²
Step 7: PT² = 144, PT = 12 cm
Final Answer: Length of tangent = 12 cm
Q2: Two tangents are drawn to a circle from an external point. Prove that they are equal in length.
Step 1: Let P be external point and PA, PB be tangents to circle with center O
Step 2: Since tangent is perpendicular to radius: OA ⊥ PA and OB ⊥ PB
Step 3: In triangles OAP and OBP:
OA = OB (radii)
∠OAP = ∠OBP = 90°
OP = OP (common side)
Step 4: By RHS (Right angle-Hypotenuse-Side) congruence: △OAP ≅ △OBP
Step 5: Therefore PA = PB
Final Answer: Two tangents from external point are equal
Q3: If two chords of a circle intersect at a point inside the circle, prove that the product of their segments are equal.
Step 1: Let chords AB and CD intersect at point P inside circle
Step 2: In triangles APD and CPB:
∠APD = ∠CPB (vertically opposite angles)
∠ADP = ∠CBP (angles in same segment)
Step 3: By AA similarity: △APD ~ △CPB
Step 4: Therefore AP/CP = PD/PB
Step 5: Cross-multiplying: AP × PB = CP × PD
Final Answer: AP × PB = CP × PD
Q4: A tangent and a secant are drawn to a circle from an external point P. If the tangent length is 8 cm and the secant cuts the circle such that external segment is 4 cm and whole secant is 9 cm, verify the relation.
Step 1: Tangent length PT = 8 cm
Step 2: External segment of secant = 4 cm
Step 3: Whole secant length = 9 cm
Step 4: Internal segment = 9 - 4 = 5 cm
Step 5: Power of point P: (tangent)² = (external segment) × (whole secant)
Step 6: 8² = 4 × 9
Step 7: 64 = 36... This doesn't match.
Step 8: Let me recalculate: If whole secant is 9, external is 4, then chord inside = 5
Step 9: PT² = 4 × 9 = 36, so PT = 6 cm (but given 8 cm)
Step 10: Let's verify: 8² = 64; should equal external × whole = ?
Final A…
Q5: From external point P, a secant PAB is drawn through circle with center O. If PA = 4 cm and AB = 5 cm (AB is chord), find the power of point P.
Step 1: Point P is external, A is first intersection, B is second intersection
Step 2: PA = 4 cm, AB = 5 cm
Step 3: PB = PA + AB = 4 + 5 = 9 cm
Step 4: Power of point P = PA × PB (product of segments)
Step 5: Power = 4 × 9 = 36
Final Answer: Power of point P = 36
Showing 5 of 8 questions — full solutions on the page.