USTET Mathematics — Geometry — Lines, Angles, Polygons, Triangles & CirclesFlash Cards

Find the area of a triangle with base 12 cm and height 8 cm.

Formula: A = ½bh Step 1: Identify values → b = 12 cm, h = 8 cm Step 2: Substitute into formula → A = ½(12)(8) Step 3: Calculate → A = ½(96) = 48 Answer: 48 cm²

A rectangle has length 15 m and width 8 m. Find its perimeter and area.

Perimeter: P = 2(l + w) Step 1: P = 2(15 + 8) = 2(23) = 46 m Area: A = lw Step 2: A = 15 × 8 = 120 m² Answers: Perimeter = 46 m, Area = 120 m²

Find the missing angle in a triangle where two angles are 65° and 45°.

Formula: Sum of angles in triangle = 180° Step 1: Let missing angle = x Step 2: Set up equation → 65° + 45° + x = 180° Step 3: Simplify → 110° + x = 180° Step 4: Solve → x = 180° - 110° = 70° Answer: 70°

Using the Pythagorean theorem, find the hypotenuse of a right triangle with legs 9 cm and 12 cm.

Formula: c² = a² + b² Step 1: Identify legs → a = 9 cm, b = 12 cm Step 2: Substitute → c² = 9² + 12² Step 3: Calculate → c² = 81 + 144 = 225 Step 4: Find square root → c = √225 = 15 Answer: 15 cm

Find the circumference of a circle with radius 7 cm. (Use π ≈ 3.14)

Formula: C = 2πr Step 1: Identify radius → r = 7 cm Step 2: Substitute → C = 2π(7) = 14π Step 3: Calculate numerical value → C = 14 × 3.14 = 43.96 Answer: 14π cm or ≈ 43.96 cm

Two parallel lines are cut by a transversal. If one angle is 110°, find its corresponding angle.

Property: Corresponding angles are equal when parallel lines are cut by a transversal Step 1: Identify the relationship → angles are corresponding Step 2: Apply property → corresponding angles are congruent Answer: 110° (corresponding angles are equal)

Find the area of a circle with diameter 14 cm.

Formula: A = πr² Step 1: Find radius → r = diameter ÷ 2 = 14 ÷ 2 = 7 cm Step 2: Substitute → A = π(7)² = π(49) = 49π Step 3: Calculate numerical value → A ≈ 49 × 3.14 = 153.86 Answer: 49π cm² or ≈ 153.86 cm²

Find the number of diagonals in a hexagon (6-sided polygon).

Formula: D = n(n-3)/2, where n = number of sides Step 1: Identify n → n = 6 (hexagon) Step 2: Substitute → D = 6(6-3)/2 = 6(3)/2 Step 3: Calculate → D = 18/2 = 9 Answer: 9 diagonals

In a 30-60-90 triangle, if the shortest side is 5 cm, find the hypotenuse.

Property: In 30-60-90 triangle, hypotenuse = 2 × shortest side Step 1: Identify shortest side → 5 cm (opposite 30° angle) Step 2: Apply property → hypotenuse = 2 × 5 = 10 cm Step 3: Verify ratio → sides are in ratio 1:√3:2 Answer: 10 cm

Find the sum of interior angles of a pentagon.

Formula: S = 180(n-2), where n = number of sides Step 1: Identify n → n = 5 (pentagon) Step 2: Substitute → S = 180(5-2) = 180(3) Step 3: Calculate → S = 540° Answer: 540°

Find the area of a rhombus with diagonals 12 cm and 16 cm.

Formula: A = ½(d₁ × d₂), where d₁ and d₂ are diagonals Step 1: Identify diagonals → d₁ = 12 cm, d₂ = 16 cm Step 2: Substitute → A = ½(12 × 16) Step 3: Calculate → A = ½(192) = 96 Answer: 96 cm²

Two complementary angles are in the ratio 2:3. Find both angles.

Property: Complementary angles sum to 90° Step 1: Let angles be 2x and 3x Step 2: Set up equation → 2x + 3x = 90° Step 3: Solve → 5x = 90°, so x = 18° Step 4: Find angles → 2x = 36°, 3x = 54° Answer: 36° and 54°

Find the radius of a circle if its area is 154 cm². (Use π = 22/7)

Formula: A = πr², so r = √(A/π) Step 1: Set up equation → 154 = (22/7)r² Step 2: Solve for r² → r² = 154 × 7/22 = 1078/22 = 49 Step 3: Find radius → r = √49 = 7 Answer: 7 cm

In an isosceles triangle, the base angles are each 65°. Find the vertex angle.

Property: Base angles of isosceles triangle are equal Step 1: Identify base angles → each is 65° Step 2: Sum of base angles → 65° + 65° = 130° Step 3: Find vertex angle → 180° - 130° = 50° Answer: 50°

When do you use the formula A = ½bh for finding area?

Use for: Any triangle when you know base and height Formula: A = ½bh Key points: - b = base (any side can be base) - h = perpendicular height to that base - Height must be perpendicular (90°) to base Example: Triangle with base 10 cm, height 6 cm → A = ½(10)(6) = 30 cm²

Find the exterior angle of a regular octagon.

Formula: Each exterior angle = 360°/n, where n = number of sides Step 1: Identify n → n = 8 (octagon) Step 2: Substitute → exterior angle = 360°/8 Step 3: Calculate → exterior angle = 45° Verify: All exterior angles sum to 360° → 8 × 45° = 360° ✓ Answer: 45°

A trapezoid has parallel sides of 8 cm and 12 cm, with height 5 cm. Find its area.

Formula: A = ½(b₁ + b₂)h, where b₁, b₂ are parallel sides Step 1: Identify values → b₁ = 8 cm, b₂ = 12 cm, h = 5 cm Step 2: Substitute → A = ½(8 + 12)(5) Step 3: Calculate → A = ½(20)(5) = ½(100) = 50 Answer: 50 cm²

Two angles are supplementary. If one angle is 3 times the other, find both angles.

Property: Supplementary angles sum to 180° Step 1: Let smaller angle = x, larger angle = 3x Step 2: Set up equation → x + 3x = 180° Step 3: Solve → 4x = 180°, so x = 45° Step 4: Find both angles → x = 45°, 3x = 135° Answer: 45° and 135°

Find the volume of a rectangular prism with length 12 cm, width 8 cm, and height 6 cm.

Formula: V = lwh Step 1: Identify dimensions → l = 12 cm, w = 8 cm, h = 6 cm Step 2: Substitute → V = 12 × 8 × 6 Step 3: Calculate → V = 96 × 6 = 576 Answer: 576 cm³

In a 45-45-90 triangle, if one leg is 8 cm, find the hypotenuse.

Property: In 45-45-90 triangle, hypotenuse = leg × √2 Step 1: Identify leg length → 8 cm Step 2: Apply property → hypotenuse = 8√2 cm Step 3: Calculate decimal → hypotenuse ≈ 8 × 1.414 = 11.31 cm Answer: 8√2 cm or ≈ 11.31 cm