Practice Test Geeks(NBT) National Benchmark Test Practice Test
← All NBT Flashcard Decks

Mathematics: Geometry and Trigonometry Flashcards

6 cards from real NBT practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Mathematics: Geometry and Trigonometry flashcards as text

  1. 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 area of the triangle with sides 13, 14, 15 is found using Heron's formula: s = (13+14+15)/2 = 21. Area = √(21×8×7×6) = √7056 = 84 cm². The inradius r = Area/s = 84/21 = 4 cm.

  2. If sin θ + cos θ = √2 · sin(θ + 45°) is an identity, what is the value of sin²θ + cos²θ + 2sinθcosθ when expressed solely in terms of a known constant?

    Answer: 1 + sin 2θ

    (sin θ + cos θ)² = sin²θ + 2sinθcosθ + cos²θ = 1 + 2sinθcosθ = 1 + sin 2θ. This is the expanded form of the square of the given expression, not simply 1 or 2.

  3. Two tangent lines are drawn from an external point P to a circle of radius 5 cm. If the distance from P to the center O is 13 cm, what is the length of each tangent segment and the area of quadrilateral OAOB (where A and B are the tangent points)?

    Answer: 12 cm; 60 cm²

    Tangent length = √(13² − 5²) = √(169 − 25) = √144 = 12 cm. Quadrilateral OAOB consists of two right triangles OAP and OBP (each with legs 5 and 12). Area of each = ½ × 5 × 12 = 30 cm², so total area = 60 cm².

  4. In triangle ABC, angle A = 30°, angle B = 45°, and side b (opposite B) = 6 cm. Using the sine rule, what is the exact length of side a (opposite A)?

    Answer: 3√2 cm

    By the sine rule: a/sin A = b/sin B → a/sin 30° = 6/sin 45°. So a = 6 × (sin 30°/sin 45°) = 6 × (½ ÷ (√2/2)) = 6 × (1/√2) = 6/√2 = 3√2 cm.

  5. A regular hexagon has a diagonal connecting two opposite vertices of length 10 cm. What is the area of the hexagon?

    Answer: 75√3 cm²

    The longest diagonal of a regular hexagon equals 2r (twice the circumradius). So 2r = 10 → r = 5 cm. The area of a regular hexagon = (3√3/2)r² = (3√3/2)(25) = 75√3/2 × 2 = 75√3 cm². Alternatively, side length s = r = 5, and area = (3√3/2)s² = 75√3/2 × 2 — the full formula gives (3√3/2)(25) = 75√3 cm².

  6. The angle of elevation to the top of a vertical tower from point A is 60°. From point B, which is 20 m further from the base of the tower along the same horizontal line as A, the angle of elevation is 30°. What is the height of the tower?

    Answer: 10√3 m

    Let h = height of tower and d = distance from A to base. Then tan 60° = h/d → h = d√3, and tan 30° = h/(d+20) → h = (d+20)/√3. Setting equal: d√3 = (d+20)/√3 → 3d = d+20 → d = 10 m. Therefore h = 10√3 m.