Geometry

Quick Answer: Geometry questions test your knowledge of shape properties, area, perimeter, volume, and basic coordinate geometry. Memorize key formulas (you won't have a formula sheet), understand angle relationships, and practice the Pythagorean theorem. These concrete skills are directly testable.

Basic Geometric Concepts

Types of Angles

Angle Relationships

Worked Example: Angle Relationships

Problem: Two angles are supplementary. One angle is 35° more than twice the other. Find both angles.

Solution:

1. Let x = smaller angle
2. Larger angle = 2x + 35
3. Supplementary: x + (2x + 35) = 180
4. Solve: 3x + 35 = 180 → 3x = 145 → x = 48.33°
5. Larger angle: 2(48.33) + 35 = 131.67°

Check: 48.33 + 131.67 = 180° ✓

Triangles

Triangle Properties

• Sum of interior angles = 180°
• Exterior angle = sum of two non-adjacent interior angles
• Sum of any two sides > third side (Triangle Inequality)

Types of Triangles

Triangle Area

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Pythagorean Theorem

For right triangles:

$a^2 + b^2 = c^2$

Where c is the hypotenuse (longest side, opposite the right angle).

Worked Example: Pythagorean Theorem

Problem: A right triangle has legs of 6 and 8. Find the hypotenuse.

Solution: $c^2 = 6^2 + 8^2 = 36 + 64 = 100$ $c = \sqrt{100} = \textbf{10}$

Calculator Tip: The on-screen calculator has a square root function. Enter 100, then click the √ button to get 10.

Common Pythagorean Triples

Memorize these for quick recognition:

• 3-4-5 (and multiples: 6-8-10, 9-12-15)
• 5-12-13
• 8-15-17

Quadrilaterals

Properties Summary

Perimeter and Area Formulas

Worked Example: Composite Shape

Problem: A rectangular classroom is 30 feet by 24 feet. A triangular reading corner with base 8 feet and height 6 feet is carpeted differently. What is the area of the non-reading-corner floor?

Solution:

1. Rectangle area: 30 × 24 = 720 sq ft
2. Triangle area: ½ × 8 × 6 = 24 sq ft
3. Non-corner area: 720 - 24 = 696 sq ft

Circles

Circle Formulas

Where r = radius, d = diameter, π ≈ 3.14159

Worked Example: Circle Calculations

Problem: A circular garden has a radius of 7 meters. Find its circumference and area. (Use π ≈ 3.14)

Solution:

• Circumference: C = 2πr = 2 × 3.14 × 7 = 43.96 meters
• Area: A = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 sq meters

Calculator Tip: For circle problems, use 3.14 for π unless the question specifies otherwise. Enter the calculation step by step: 3.14 × 7 × 7 = for the area.

Volume

Volume Formulas

Worked Example: Volume

Problem: A rectangular aquarium is 40 cm long, 25 cm wide, and 30 cm tall. What is its volume?

Solution: V = 40 × 25 × 30 = 30,000 cubic centimeters (or 30 liters)

Coordinate Geometry

The Coordinate Plane

Distance Formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Worked Example: Distance

Problem: Find the distance between (1, 2) and (4, 6).

Solution: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = \textbf{5}$

Midpoint Formula

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Worked Example: Midpoint

Problem: Find the midpoint of the segment connecting (2, 8) and (6, 2).

Solution: $\text{Midpoint} = \left(\frac{2 + 6}{2}, \frac{8 + 2}{2}\right) = \left(\frac{8}{2}, \frac{10}{2}\right) = \textbf{(4, 5)}$

Transformations (Basic Concepts)

Test Tip: When working with geometric figures on a coordinate plane, sketch a quick diagram on your scratch paper. Visual representation helps avoid errors and clarifies the problem.