Geometric Constraints Examples in Math

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: Constraints reduce degrees of freedom and define geometric relationships.

Common stuck point: Finding what constraints imply about possible configurations.

Sense of Study hint: List every condition the problem gives you. Then ask: with these conditions, what shapes or positions are still possible?

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.

Solution

  1. 1
    Step 1: Circle constraint: x^2 + y^2 = 25. Line constraint: y = x + 1.
  2. 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. 3
    Step 3: Factor: (x+4)(x-3) = 0, so x = -4 or x = 3.
  4. 4
    Step 4: Corresponding y-values: y = -4+1 = -3 and y = 3+1 = 4. Points: (-4, -3) and (3, 4).

Answer

P = (-4, -3) or P = (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.

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.
← Back to Geometric Constraints Practice Mode

Background Knowledge

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

shapes