Tangents and Secants to a Circle — Telangana (SSC) Class 10 Mathematics Solutions (Free)

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

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

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

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 Answer: Using tangent-secant formula: (tangent)² = (near distance) × (far distance) from external point

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

Step 1: For a tangent to a circle, the distance from center to tangent must equal the radius Step 2: Radius = 6 cm Step 3: Distance from center to tangent = 6 cm Step 4: Distance equals radius Step 5: This is exactly the condition for a tangent Step 6: The tangent touches the circle at exactly one point Final Answer: Yes, this is possible. The line is tangent to the circle.

Step 1: Radius of first circle r₁ = 5 cm, radius of second r₂ = 3 cm Step 2: Distance between centers d = 10 cm Step 3: Check relationships: If d > r₁ + r₂: circles are external (4 tangents) If d = r₁ + r₂: circles touch externally (3 tangents) If |r₁ - r₂| < d < r₁ + r₂: circles intersect (2 tangents) If d = |r₁ - r₂|: circles touch internally (1 tangent) If d < |r₁ - r₂|: one circle inside other (0 tangents) Step 4: r₁ + r₂ = 5 + 3 = 8 cm; |r₁ - r₂| = 2 cm Step 5: Since 8 < 10, we have d > r₁ + r₂ Step 6: Circles are external Final Answer: 4 common tangents can be drawn

Step 1: Distance from center O to chord = d = 3 cm Step 2: Radius r = 5 cm Step 3: The perpendicular from center bisects the chord Step 4: Let M be midpoint of chord, C be one endpoint Step 5: In right triangle OMC: OM = 3, OC = 5 Step 6: MC² = OC² - OM² = 5² - 3² = 25 - 9 = 16 Step 7: MC = 4 cm Step 8: Chord length = 2 × MC = 2 × 4 = 8 cm Final Answer: Length of chord = 8 cm