Honors Geometry Summer Review Packet 2018

Purpose of This Packet

Honors Geometry places a strong emphasis on logical reasoning, proof-writing, and problem-solving. While we will study many new geometric concepts, you will rely extensively on your algebraic skills to solve geometry problems. This packet reviews essential topics including:

• Solving linear equations
• Working with systems of equations
• Simplifying algebraic expressions
• Factoring polynomials
• Solving quadratic equations
• Working with radicals
• Solving word problems
• Graphing linear equations

Topic 1: Solving Linear Equations

In geometry, you'll often need to solve equations to find missing values such as angle measures, side lengths, and coordinates. Review the following procedure for solving linear equations:

1. Remove parentheses by distributing
2. Combine like terms on each side of the equation
3. Isolate the variable term by adding or subtracting
4. Isolate the variable by multiplying or dividing
5. Check your answer by substituting back into the original equation

Answers:

1. x = 8
2. x = 8
3. x = 6
4. x = 18

Topic 2: Systems of Equations

Systems of equations will be used frequently in Honors Geometry, especially when working with coordinates, parallel lines, and solving for multiple unknown values.

Methods for Solving Systems of Equations:

• Graphing: Plot both equations on the same coordinate plane; the intersection point is the solution.
• Substitution: Solve one equation for a variable and substitute into the other equation.
• Elimination: Add or subtract equations to eliminate one variable.

Answers:

1. (5, 7)
2. (4, 2)
3. (1, 2)

Topic 3: Factoring Polynomials

Factoring skills are essential when working with area problems, quadratic equations, and many geometric situations.

Factoring Methods:

• Greatest Common Factor (GCF): Identify the largest factor common to all terms and factor it out.
• Difference of Squares: a - b = (a - b)(a + b)
• Trinomials: For ax + bx + c, find two numbers that multiply to ac and add to b
• Perfect Square Trinomials: a 2ab + b = (a b)

Answers:

1. 6x(x + 2)
2. (x - 3)(x + 3)
3. (x - 7)(x + 2)
4. (x + 6)
5. (2x + 1)(x - 3)

Topic 4: Quadratic Equations

You will encounter quadratic equations when working with the Pythagorean theorem, area problems, and coordinate geometry.

Methods for Solving Quadratic Equations:

For a quadratic equation in standard form: ax + bx + c = 0

Answers:

1. x = 0 or x = 5
2. x = 3 or x = -3
3. x = 2 or x = 4
4. x = 1 or x = -5
5. x = 2 or x = 0.5

Topic 5: Radicals and Radical Equations

Working with radicals (square roots) is essential for the Pythagorean theorem, special right triangles, and distance formula applications.

Properties of Radicals:

Simplifying Radicals:

Simplify radicals by identifying perfect square factors:

Answers:

1. 32
2. 53
3. 35
4. x = 36
5. x = 22

Topic 6: Linear Equations and Graphing

Understanding linear equations and how to graph them is fundamental to coordinate geometry and working with lines in geometric contexts.

Forms of Linear Equations:

Working with Slope:

Parallel and Perpendicular Lines:

• Parallel lines have equal slopes: m = m
• Perpendicular lines have slopes that are negative reciprocals: m m = -1

Answers:

1. 2
2. y = 2x - 1
3. -1/3
4. -2/3
5. (Line with y-intercept at -3 and slope of 2)

Topic 7: Word Problems and Problem Solving

Geometry problems are often presented as word problems. Follow these strategies:

1. Read the problem carefully and identify what you're being asked to find.
2. Draw a diagram to represent the situation when possible.
3. Identify the given information and what you need to determine.
4. Write an equation or system of equations that represents the relationship between quantities.
5. Solve the equation and check your answer in the context of the problem.
6. Write a complete sentence answering the specific question asked.

Answers:

1. Width = 6 cm, Length = 15 cm
2. 4 and 14
3. 9 units and 12 units

Important Formulas to Remember

These formulas will be used throughout Honors Geometry:

Area Formulas:

Perimeter and Circumference Formulas:

The Pythagorean Theorem:

Additional Resources

For additional help with these topics, you may consult:

• Your algebra textbook
• Khan Academy (www.khanacademy.org)
• PatrickJMT on YouTube
• Sparknotes Math Study Guides
• Paul's Online Math Notes
• Your local library for algebra review books

If you encounter persistent difficulties with any topic, make note of these so we can address them early in the school year.

Expectations for Honors Geometry

This summer review packet serves as your first introduction to the expectations of Honors Geometry. Throughout the course, you will be expected to:

• Apply algebraic skills to solve geometric problems
• Write clear, logical proofs showing your reasoning
• Communicate mathematical ideas precisely
• Work with both 2D and 3D geometric concepts
• Use technology appropriately to explore geometric relationships
• Collaborate with peers to solve complex problems
• Persevere through challenging problems that may require multiple attempts
• Connect geometric concepts to real-world applications