Analytic Geometry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Analytic Geometry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Analytic geometry studies geometric objects using coordinate systems and algebraic equations, translating shapes into formulas so that algebra can solve geometry problems. This field, founded by Descartes, unifies algebra and geometry.

It translates shapes into equations so algebra can solve geometry problems.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Analytic geometry turns a geometric object into an equation in $x$ and $y$ so you can compute instead of construct.

Common stuck point: The procedure for analytic geometry is the easy part; the trap is picking awkward coordinates that bury the algebra in fractions. Asking "Can I answer this by assigning coordinates and computing with an equation instead of drawing or proving by congruence?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I answer this by assigning coordinates and computing with an equation instead of drawing or proving by congruence?

Worked Examples

Example 1

medium

Prove that the triangle with vertices

A(0,0)

B(4,0)

, and

C(2,2\sqrt{3})

is equilateral using coordinates.

Equilateral triangle — all three sides = 4

Answer

The triangle is equilateral with all sides equal to

4

First step

Compute

AB

AB = \sqrt{(4-0)^2 + (0-0)^2} = \sqrt{16} = 4

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Example 2

hard

Use coordinates to prove that the diagonals of a rectangle bisect each other.

Example 3

medium

Find the equation of the line through

(1, -2)

and

(3, 4)

Line through (1,−2) and (3,4): slope = 3, equation y = 3x − 5.

Example 4

medium

Find the midpoint of segment

AB

where

A(-3, 7)

and

B(5, -1)

Find the midpoint M of segment AB.

Example 5

hard

Find the equation of the circle through

(0,0)

(6, 0)

, and

(0, 8)

Circle through (0,0),(6,0),(0,8): hypotenuse is diameter, center (3,4), radius 5.

Example 6

hard

Verify that the quadrilateral with vertices

A(0,0)

B(4,0)

C(4,3)

D(0,3)

is a rectangle and find its diagonal.

Rectangle with sides 4 and 3: diagonal = √(16+9) = 5.

Example 7

challenge

Using coordinates, prove that the midpoints of the sides of any rectangle form a rhombus. Take vertices

A(0,0)

B(2a, 0)

C(2a, 2b)

D(0, 2b)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Determine whether points

P(1,2)

Q(3,6)

, and

R(5,10)

are collinear using the slope method.

Are P, Q, R collinear? Compare slopes PQ and QR.

Example 2

medium

Show that quadrilateral

A(1,1)

B(4,2)

C(5,5)

D(2,4)

is a parallelogram by comparing slopes of opposite sides.

Is ABCD a parallelogram? Check slopes of opposite sides.

Example 3

easy

Find the distance between

A(1, 2)

and

B(4, 6)

Find the distance d between A(1,2) and B(4,6).

Example 4

easy

Write the equation of a line with slope

4

and y-intercept

-3

Example 5

medium

Find the equation of the line through

(2, 5)

with slope

-3

in slope-intercept form.

Example 6

medium

Find the equation of the circle centered at

(3, -1)

with radius

4

Example 7

medium

Find the slope of a line perpendicular to

y = \dfrac{2}{3}x + 5

Example 8

medium

Find the slope of a line parallel to

4x - 2y = 7

Example 9

medium

Does the point

(3, 4)

lie on the circle

x^2 + y^2 = 25

Example 10

medium

The vertices of a triangle are

A(0,0)

B(6,0)

C(0,8)

. Find its area.

Area of triangle A(0,0), B(6,0), C(0,8) = ½ × 6 × 8 = 24.

Example 11

hard

Show that the triangle with vertices

A(1, 1)

B(4, 5)

C(8, 2)

is a right triangle.

Right angle at B: slopes 4/3 and −3/4 multiply to −1.

Example 12

hard

Find the equation of the perpendicular bisector of segment

AB

where

A(2, 1)

and

B(8, 9)

Example 13

hard

Rewrite the circle equation

x^2 + y^2 - 6x + 4y - 12 = 0

in standard form and find its center and radius.

Example 14

hard

Find the area of the triangle with vertices

A(1, 2)

B(5, 2)

C(3, 8)

Area = ½ × base × height = ½ × 4 × 6 = 12.

Example 15

hard

Find the equation of the line that passes through

(4, -1)

and is parallel to

3x + 2y = 8

Example 16

challenge

Find the intersection of the lines

y = 2x + 1

and

y = -x + 7

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Related Concepts

Coordinate Representation Distance Formula Slope in Geometry

Background Knowledge

These ideas may be useful before you work through the harder examples.

coordinate representationdistance formulaslope in geometry