Tangent Lines

Step 1: Understand the tangent property: A tangent line is perpendicular to the radius at the point of tangency

Step 2: Identify the right triangle: Triangle OPQ has:

• Right angle at P (tangent ⊥ radius)
• OP = 5 cm (radius)
• OQ = 13 cm (given distance)
• PQ = ? (distance along tangent)

Step 3: Use Pythagorean Theorem: OP² + PQ² = OQ² 5² + PQ² = 13² 25 + PQ² = 169

Step 4: Solve for PQ: PQ² = 169 - 25 PQ² = 144 PQ = 12 cm

Step 5: Recognize the triple: This is a 5-12-13 right triangle

Answer: PQ = 12 cm

Step 1: Recall the Two-Tangent Theorem: Tangent segments drawn from an external point to a circle are congruent (equal in length)

Step 2: Apply the theorem: If one tangent segment = 8 cm Then the other tangent segment = 8 cm

Step 3: Understand why: Both segments are from the same external point to points of tangency on the circle By symmetry and congruent triangles, they're equal

Answer: The other tangent segment is 8 cm

Step 1: Draw and understand the figure:

• P is external point
• PA and PB are tangent segments (PA = PB = 15)
• O is the center
• Radius OA = OB = 9

Step 2: Identify the right triangle: Triangle OAP has:

• Right angle at A (tangent ⊥ radius)
• OA = 9 cm (radius)
• PA = 15 cm (tangent segment)
• OP = ? (distance to center)

Step 3: Use Pythagorean Theorem: OA² + PA² = OP² 9² + 15² = OP² 81 + 225 = OP² 306 = OP²

Step 4: Solve for OP: OP = √306 OP = √(9 × 34) OP = 3√34 cm

Step 5: Approximate: OP ≈ 3 × 5.831 ≈ 17.49 cm

Answer: OP = 3√34 cm (≈ 17.49 cm)

Step 1: Understand the setup: Two circles with radii r₁ = 8 cm and r₂ = 12 cm Distance between centers = 25 cm Need length of external tangent segment

Step 2: Use the external tangent formula: For external tangent between circles: Create a right triangle by drawing a line parallel to the tangent from the smaller circle's center

Step 3: Find the height of the right triangle: Height = |r₂ - r₁| = |12 - 8| = 4 cm

Step 4: The hypotenuse is the distance between centers: Hypotenuse = 25 cm

Step 5: Use Pythagorean Theorem: Tangent length² + 4² = 25² Tangent length² + 16 = 625 Tangent length² = 609 Tangent length = √609 cm

Step 6: Simplify: √609 = √(21 × 29) (cannot simplify further) √609 ≈ 24.68 cm

Step 7: Total belt length consideration: The question asks for ONE external tangent segment Length = √609 ≈ 24.68 cm

Answer: The external tangent length is √609 cm (≈ 24.68 cm)

Step 1: Understand the configuration:

• PA and PB are equal tangent segments = 10 cm
• Angle APB = 60°
• Need to find radius and PO

Step 2: Use symmetry: Since PA = PB, triangle PAB is isosceles PO bisects angle APB So angle APO = angle BPO = 30°

Step 3: Analyze triangle OAP:

• Angle OAP = 90° (tangent ⊥ radius)
• Angle APO = 30°
• Therefore angle AOP = 60° (angles sum to 180°)

This is a 30-60-90 triangle!

Step 4: Use 30-60-90 triangle ratios: In a 30-60-90 triangle: sides are in ratio 1 : √3 : 2

• Side opposite 30° (OA = radius) : shortest
• Side opposite 60° (PA = 10) : middle
• Side opposite 90° (PO) : longest

Step 5: Find the radius: If PA (opposite 60°) = 10 = x√3 Then x = 10/√3 = 10√3/3 Radius OA = x = 10√3/3 cm

Step 6: Find PO: PO (opposite 90°) = 2x = 2(10√3/3) = 20√3/3 cm

Step 7: Alternative method using right triangle: In triangle OAP: tan(30°) = OA/PA OA = PA × tan(30°) = 10 × (1/√3) = 10/√3 = 10√3/3 ✓

cos(30°) = PA/PO PO = PA/cos(30°) = 10/(√3/2) = 20/√3 = 20√3/3 ✓

Step 8: Approximate values: Radius ≈ 10(1.732)/3 ≈ 5.77 cm PO ≈ 20(1.732)/3 ≈ 11.55 cm

Answer: (a) Radius = 10√3/3 cm (≈ 5.77 cm) (b) Distance PO = 20√3/3 cm (≈ 11.55 cm)