Geometry — Lines, Angles, Polygons, Triangles & Circles - Flashcards
This comprehensive flashcard set covers essential geometry concepts for UPCAT preparation. Focus on problem-solving skills, formula applications, and step-by-step solutions. These cards emphasize practical problem-solving over theoretical definitions, helping you master geometry through worked examples and real applications.
Flashcards
Find the area of a triangle with base 12 cm and height 8 cm.
Answer
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.
Answer
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°.
Answer
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.
Answer
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)
Answer
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.
Answer
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.
Answer
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).
Answer
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.
Answer
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.
Answer
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.
Answer
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.
Answer
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)
Answer
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.
Answer
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?
Answer
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.
Answer
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.
Answer
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.
Answer
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.
Answer
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.
Answer
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
Diagrams
Geometry Concepts Overview
Comprehensive overview of all geometry topics covered in the flashcards, showing the hierarchical relationship between concepts.
Geometry Problem-Solving Decision Tree
Step-by-step decision tree to help students choose the right formula and approach for different geometry problems.