Geometric Constraints Examples: 45 Problems with Answers

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

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

Concept Recap

Conditions that limit or restrict the possible positions, sizes, or shapes of geometric objects in a problem.

A door hinge constrains the door to swing in an arc, not slide sideways.

Read the full concept explanation →

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: A geometric constraint is a condition that limits where a figure's points can go or what sizes it can take.

Common stuck point: The procedure for geometric constraints is the easy part; the trap is treating one constraint as enough to fix a figure. Asking "Is this a rule that limits where points can go or what sizes are allowed, rather than a single answer?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a rule that limits where points can go or what sizes are allowed, rather than a single answer?

Worked Examples

Example 1

easy

A point

P

is constrained to lie on a circle of radius

5

centred at the origin AND on the line

y = x + 1

. Find all possible positions of

P

Answer

P = (-4, -3)

P = (3, 4)

First step

Step 1: Circle constraint:

x^2 + y^2 = 25

. Line constraint:

y = x + 1

Full solution

2
Step 2: Substitute $y = x+1$ into the circle: $x^2 + (x+1)^2 = 25 \Rightarrow x^2 + x^2 + 2x + 1 = 25 \Rightarrow 2x^2 + 2x - 24 = 0 \Rightarrow x^2 + x - 12 = 0$ .
3
Step 3: Factor: $(x+4)(x-3) = 0$ , so $x = -4$ or $x = 3$ .
4
Step 4: Corresponding $y$ -values: $y = -4+1 = -3$ and $y = 3+1 = 4$ . Points: $(-4, -3)$ and $(3, 4)$ .

Geometric constraints reduce the infinite set of possible positions to a finite (or smaller) set. Here two constraints — a circle and a line — together allow only two solutions, found by solving the system of equations.

Example 2

medium

A rectangle has perimeter

36

cm and one side of length

x

. Write the constraint for the other side, and determine the range of valid values for

x

Example 3

easy

Write the constraints describing the closed disk of radius

4

centered at

(1, -2)

Example 4

medium

Point

P

must satisfy: (i)

P

is on the line

y = 2x - 1

, (ii)

P

is exactly distance

1

from the origin. Find all

P

Example 5

medium

A point lies on both the circle

x^2 + y^2 = 25

and the circle

(x - 6)^2 + y^2 = 25

. Find both intersection points.

Example 6

medium

Find all points equidistant from

A(0, 0)

and

B(6, 0)

AND on the circle

x^2 + y^2 = 25

Example 7

medium

Suppose a rectangle has perimeter

40

cm. Write the area as a function of one side

x

and find the range of

x

Example 8

hard

Find the set of points

P(x, y)

such that the distance from

P

(0, 2)

is twice the distance from

P

(0, -1)

Example 9

hard

Find the locus of points

P

such that

PA^2 + PB^2 = 50

, where

A(-3, 0)

and

B(3, 0)

Example 10

hard

Find the set of points

P

such that the angle

\angle APB = 90°

where

A(-5, 0)

and

B(5, 0)

Example 11

challenge

Five points in the plane are positioned so that every pair is distance at least

1

apart. Show that not all five points can fit inside a closed unit square.

Practice Problems

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

Example 1

easy

A point

P(x, y)

must satisfy: (1) it is in the first quadrant, and (2) its distance from the origin is at most

4

. Write the constraints as inequalities.

Example 2

hard

A triangle has sides

a

b

c

with perimeter

30

. Write all the geometric constraints (triangle inequality and perimeter) that

a

b

c

must satisfy, then find the maximum possible value of side

a

given

b = c

Example 3

easy

The set of points exactly 3 units from a fixed point

P

forms what shape?

Example 4

easy

The set of points equidistant from two points

A

and

B

forms what?

Example 5

easy

A triangle has angles that must sum to a fixed value. What is that constraint?

Example 6

easy

The points at distance 2 from a fixed line form what?

Example 7

easy

A door on a hinge can only swing. What does the hinge constrain the door to move along?

Example 8

easy

If three side lengths are fixed, how many differently-shaped triangles can be built (ignoring position)?

Example 9

easy

A point must lie on BOTH a given circle and a given line. How many positions can it have (at most)?

Example 10

easy

Can a triangle have sides 3, 3, and 10? What constraint does this violate?

Example 11

medium

A point is equidistant from

A

and

B

AND lies on a given line. How many such points are there (typically)?

Example 12

medium

Why does fixing two angles of a triangle automatically fix the third?

Example 13

medium

A point is 5 from the origin and 5 from

(8, 0)

. Find its possible positions.

Example 14

medium

A rectangle has a fixed perimeter of 20. What single constraint relates its length

l

and width

w

Example 15

medium

What does it mean to 'over-constrain' a figure? Give an example.

Example 16

medium

A point lies on the line

y = x

and is 5 units from the origin. Find its possible positions.

Example 17

medium

How many points are equidistant from all three vertices of a triangle, and what is that point called?

Example 18

medium

A ladder of fixed length leans with its base on the floor and top on a wall. As the base slides out, what constrains the top?

Example 19

challenge

A point must be 3 from line

y = 0

and 4 from the point

(0, 0)

. Find all positions.

Example 20

challenge

In the plane, how many degrees of freedom does a single point have, and how many does a constraint 'lies on a given line' remove?

Example 21

challenge

A point is equidistant from three given points that happen to be collinear. How many such points exist?

Example 22

challenge

Explain why a four-bar linkage (four rigid rods hinged in a loop) can move, but a triangle of three rods cannot.

Example 23

easy

A point

P(x, y)

lies inside the rectangle with corners

(0, 0)

and

(8, 5)

. Write the constraints as inequalities.

Example 24

easy

Can a triangle have sides

2

5

8

? Why or why not?

Example 25

easy

A square has perimeter

32

cm. Write the constraint on its side length

s

Example 26

easy

A rectangle has area

48

sq cm and one side

x

. Write the other side in terms of

x

and state the constraint on

x

Example 27

medium

Three rods of lengths

a

b

c

are joined to form a triangle. The constraint

|b - c| < a < b + c

is called the ___.

Example 28

medium

An angle in a triangle must be greater than

0°

and less than

180°

, and the three must sum to

180°

. How many free parameters remain after these constraints (for the angle triple)?

Example 29

medium

A right triangle has legs

a

and

b

and hypotenuse

10

. Write the constraint on

a

and

b

Example 30

medium

A line passes through

(0, 0)

and is tangent to the circle

(x - 6)^2 + y^2 = 9

. Find the absolute value of its slope.

Example 31

medium

How many lines pass through a given external point and are tangent to a given circle?

Example 32

hard

A triangle has sides

5

x

12

. List all integer values of

x

that satisfy the triangle inequality.

Example 33

hard

A rectangle has integer side lengths and area

36

. List the possible perimeters.

Example 34

hard

A point

P

must satisfy: (i)

x \geq 0

, (ii)

y \geq 0

, (iii)

x + y \leq 10

. Find the maximum value of

x \cdot y

subject to these constraints.

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Related Concepts

Basic Shapes Geometric Constraints

Background Knowledge

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

shapes