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Analytic Geometry · Analytic Geometry

Analytic Geometry: Unit Review

1.1 1.2 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3About 40 minutes
Learning goals
  • I can solve linear systems by graphing, substitution and elimination, and use them to solve problems.
  • I can use the midpoint and length formulas and the equation x² + y² = r².
  • I can find equations of right bisectors, medians and altitudes, and the distance from a point to a line.
  • I can investigate and verify properties of triangles and quadrilaterals with analytic geometry.

Words to know

linear system
Two linear equations considered together; the solution makes both true.
midpoint formula
M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
length formula
AB = √[(x₂ − x₁)² + (y₂ − y₁)²].
circle centred at the origin
x² + y² = r², where r is the radius.
right bisector
The line perpendicular to a segment through its midpoint.
midsegment
A segment joining the midpoints of two sides of a triangle; parallel to the third side and half its length.

Let's review!

In this unit you solved linear systems by graphing, substitution and elimination, and used them to model real problems: tickets, salaries, currents, mixtures and investments. You developed and used the formulas for the midpoint and length of a line segment and the equation of a circle centred at the origin. You combined slopes, midpoints and lengths to find right bisectors, medians, altitudes and the distance from a point to a line. Finally, you investigated and verified properties of triangles and quadrilaterals with analytic geometry, including multi-step verifications and general proofs.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do two or three sections at a time rather than all at once. Write your full solution before checking, and always check answers to systems in both equations. For every question you miss, write down why (arithmetic slip? wrong formula? did not know where to start?) and reread that lesson's worked examples. Keep graph paper handy, and use GeoGebra or Desmos to check graphs if you have a device. Finish with the 10-question quiz.

MidpointM = ((x₁ + x₂)/2, (y₁ + y₂)/2)LengthAB = √[(x₂ − x₁)² + (y₂ − y₁)²]Slopem = (y₂ − y₁)/(x₂ − x₁)Parallel / perpendicularequal slopes / product −1Circle, centre (0, 0)x² + y² = r²Linear systemgraph, substitute or eliminate
The key formulas and methods of the unit on one card. Copy it onto an index card for studying.
xyA(−2, −1)B(4, 3)M(1, 1)Δx = 6Δy = 4AB = √52 ≈ 7.21slope AB = 4/6 = 2/3right bisector: y = −1.5x + 2.5
One segment, many tools: for A(−2, −1) and B(4, 3), the run and rise give the length √52 and slope 2/3; the midpoint M(1, 1) and the perpendicular slope −3/2 give the dashed right bisector.
Just for fun

Why did the analytic geometry student bring graph paper to the party?

She heard there would be lots of points of interest, and she wanted to plot her course.

Linear Systems: Solving by Graphing

1

Is (3, −2) a solution of 2x + y = 4 and x − y = 5?

2

Solve by graphing: y = x − 1 and y = −2x + 5.

3

How many solutions: y = 3x + 2 and y = 3x − 1?

Solving Linear Systems by Substitution

1

Solve: y = 2x − 1 and 3x + 2y = 12.

2

Solve: x − 2y = 4 and 3x + y = 5.

3

Substitution gives 0 = 3. What does that mean?

Solving Linear Systems by Elimination

1

Solve: 2x + 3y = 7 and 4x − 3y = −1.

2

Solve: 3x + 2y = 1 and 5x + 3y = 3.

3

Solve: 0.4x + 0.1y = 1.1 and 0.2x − 0.3y = −1.9.

Choosing a Method and Predicting the Number of Solutions

1

Which method is best for y = 5x and 2x + y = 14? Solve it.

2

How many solutions: 4x − 2y = 6 and 6x − 3y = 9?

3

Find k so that y = kx + 1 and 3x − y = 4 have no solution.

Modelling Problems with Linear Systems

1

A school sold 400 tickets: students $5, adults $9. Revenue was $2560. How many adult tickets?

2

Plan A: $50 + $0.20/km. Plan B: $30 + $0.30/km. When are they equal?

3

A rectangle's perimeter is 46 m and its length is 7 m more than its width. Find its dimensions.

Rate, Mixture and Investment Problems

1

A boat goes 36 km downstream in 2 h and back in 3 h. Find the boat's speed and the current.

2

How many litres of 15% and 40% solution make 20 L of 25% solution?

3

$4000 is split between 2% and 6% accounts, earning $180 in a year. How much is in each?

The Midpoint of a Line Segment

1

Find the midpoint of (−5, 8) and (3, −2).

2

M(2, −1) is the midpoint of AB with A(−3, 4). Find B.

3

Find the midpoints of the sides of the triangle (0, 0), (6, 4), (2, −8).

The Length of a Line Segment

1

Find the length from (−2, 3) to (4, −5).

2

Find the exact length from (1, 1) to (4, 7).

3

Which is closer to (0, 0): (5, 5) or (−7, 1)?

The Equation of a Circle Centred at the Origin

1

Write the equation of the circle with centre (0, 0) and radius 11.

2

Find the radius of x² + y² = 45.

3

Is (−6, 7) on, inside or outside x² + y² = 81?

4

Find the equation of the circle centred at the origin through (8, −15).

Equations of Right Bisectors, Medians and Altitudes

1

Find the right bisector of the segment from (−2, −1) to (4, 3).

2

Triangle A(0, 4), B(−2, −2), C(6, 0). Find the median from A.

3

Same triangle: find the altitude from A.

The Distance from a Point to a Line

1

Find the distance from (2, −3) to y = 4.

2

Find the distance from (5, 0) to y = x + 1.

3

Why is the perpendicular distance the shortest?

Investigating Properties of Triangles and Quadrilaterals

1

Find the centroid of (−3, 1), (5, 2), (1, 6).

2

Give a counterexample to: "Every parallelogram has perpendicular diagonals."

3

Squares on the legs of a right triangle have areas 64 and 225. What is the area of the square on the hypotenuse?

Verifying Properties of Lines and Triangles

1

Are the lines through (0, 0), (2, 5) and through (1, 1), (6, −1) perpendicular?

2

Classify triangle (0, 0), (4, 0), (2, 5) by its sides.

3

Verify that (1, 1), (4, 5), (8, 2) is a right triangle.

Verifying Properties of Quadrilaterals

1

Show that (0, 0), (4, 1), (5, 5), (1, 4) is a rhombus.

2

Is it a square?

3

Verify that the diagonals of (0, 0), (4, 1), (5, 5), (1, 4) are perpendicular.

Multi-Step Strategies: Verifying Geometric Properties

1

Triangle A(0, 0), B(4, 6), C(8, 2). M, N are midpoints of AB, AC. Verify MN ∥ BC and MN = ½BC.

2

Find the circumcentre of (0, 0), (10, 0), (0, 4).

3

Write the plan (no calculations) to verify that a quadrilateral is a rectangle.

Check your understanding

Quiz

1.

What is the solution of x + 2y = 7 and 3x − 2y = 5?

2.

Which system has infinitely many solutions?

3.

A canoe goes downstream at 9 km/h and upstream at 5 km/h. What is the current's speed?

4.

What is the midpoint of (−6, 4) and (2, −8)?

5.

What is the length of the segment from (−1, 2) to (5, 10)?

6.

What is the radius of x² + y² = 64?

7.

The slope of AB is 4/5. The slope of its right bisector is:

8.

Which steps find the distance from a point to a line?

9.

Diagonals of a quadrilateral have the same midpoint and equal lengths, and are not perpendicular. It is a:

10.

In triangle ABC, M and N are the midpoints of AB and AC, and BC = 14. Which is true?

For parents and teachers

How to use the review results: match mistakes to lessons.

  • Errors checking or graphing systems, or reading the crossing point: Linear Systems: Solving by Graphing.
  • Missing brackets or sign errors when substituting: Solving Linear Systems by Substitution.
  • Multiplying only part of an equation, or adding when you should subtract: Solving Linear Systems by Elimination.
  • Unsure which method to use, or how many solutions a system has: Choosing a Method and Predicting the Number of Solutions.
  • Trouble writing the equations for a word problem: Modelling Problems with Linear Systems; for currents, mixtures and interest: Rate, Mixture and Investment Problems.
  • Subtracting instead of averaging for a midpoint, or finding a missing endpoint: The Midpoint of a Line Segment.
  • Errors with squares and square roots in lengths: The Length of a Line Segment.
  • Confusing r and r²: The Equation of a Circle Centred at the Origin.
  • Using the wrong slope or point for a right bisector, median or altitude: Equations of Right Bisectors, Medians and Altitudes.
  • Measuring to the wrong point on a line: The Distance from a Point to a Line.
  • Believing one example proves a rule: Investigating Properties of Triangles and Quadrilaterals.
  • Choosing slopes vs lengths, or missing a concluding statement: Verifying Properties of Lines and Triangles and Verifying Properties of Quadrilaterals.
  • Getting lost in long problems: Multi-Step Strategies (write the plan first).