Mathematics: Geometry and Trigonometry Flashcards

A circle is inscribed in a triangle with sides 13 cm, 14 cm, and 15 cm. What is the radius of the inscribed circle?

Answer: 4 cm

The semi-perimeter is s = (13+14+15)/2 = 21. The area by Heron's formula: A = √(21·8·7·6) = √7056 = 84. The inradius r = A/s = 84/21 = 4 cm.

If sin θ + cos θ = √2 · sin(θ + 45°), which of the following identities confirms this result?

Answer: sin(A+B) = sinA cosB + cosA sinB

Expanding √2·sin(θ+45°) = √2[sinθ·cos45° + cosθ·sin45°] = √2[sinθ·(1/√2) + cosθ·(1/√2)] = sinθ + cosθ. This uses the compound angle identity sin(A+B) = sinA cosB + cosA sinB.

Two tangents are drawn from an external point P to a circle of radius 5 cm. The distance from P to the centre O is 13 cm. What is the length of each tangent?

Answer: 12 cm

A tangent is perpendicular to the radius at the point of contact, forming a right angle. By Pythagoras: tangent² = PO² − r² = 13² − 5² = 169 − 25 = 144, so tangent = 12 cm.

In triangle ABC, a = 7, b = 8, and angle C = 60°. Using the cosine rule, what is the length of side c?

Answer: √57

By the cosine rule: c² = a² + b² − 2ab·cosC = 49 + 64 − 2(7)(8)·cos60° = 113 − 112·(0.5) = 113 − 56 = 57. Therefore c = √57.

A sector of a circle has a perimeter of 30 cm and an arc length of 14 cm. What is the area of the sector?

Answer: 56 cm²

The perimeter of a sector = arc + 2r. So 30 = 14 + 2r → r = 8 cm. Area of sector = ½ × r × arc length = ½ × 8 × 14 = 56 cm².

Point P lies on chord AB of a circle such that AP = 4 cm and PB = 9 cm. A second chord CD passes through P with CP = 3 cm. What is the length of PD?

Answer: 12 cm

By the intersecting chords theorem: AP × PB = CP × PD → 4 × 9 = 3 × PD → 36 = 3·PD → PD = 12 cm.