Practice Geometric Constraints in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

hard
Find the set of points PP such that the angle ∠APB=90°\angle APB = 90° where A(−5,0)A(-5, 0) and B(5,0)B(5, 0).

Example 2

medium
A line passes through (0,0)(0, 0) and is tangent to the circle (x−6)2+y2=9(x - 6)^2 + y^2 = 9. Find the absolute value of its slope.

Example 3

hard
A triangle has sides 55, xx, 1212. List all integer values of xx that satisfy the triangle inequality.

Example 4

medium
A point lies on both the circle x2+y2=25x^2 + y^2 = 25 and the circle (x−6)2+y2=25(x - 6)^2 + y^2 = 25. Find both intersection points.

Example 5

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

Example 6

hard
Find the locus of points PP such that PA2+PB2=50PA^2 + PB^2 = 50, where A(−3,0)A(-3, 0) and B(3,0)B(3, 0).

Example 7

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

Example 8

medium
A point is 5 from the origin and 5 from (8,0)(8, 0). Find its possible positions.

Example 9

challenge
Five points in the plane are positioned so that every pair is distance at least 11 apart. Show that not all five points can fit inside a closed unit square.

Example 10

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

Example 11

hard
Find the set of points P(x,y)P(x, y) such that the distance from PP to (0,2)(0, 2) is twice the distance from PP to (0,−1)(0, -1).

Example 12

medium
Find all points equidistant from A(0,0)A(0, 0) and B(6,0)B(6, 0) AND on the circle x2+y2=25x^2 + y^2 = 25.

Example 13

hard
A rectangle has integer side lengths and area 3636. List the possible perimeters.

Example 14

medium
A rectangle has a fixed perimeter of 20. What single constraint relates its length ll and width ww?

Example 15

easy
True or false: specifying two sides and one angle of a triangle always determines a unique triangle.

Example 16

easy
A point P(x,y)P(x, y) must satisfy: (1) it is in the first quadrant, and (2) its distance from the origin is at most 44. Write the constraints as inequalities.

Example 17

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 18

easy
The set of points exactly 3 units from a fixed point PP forms what shape?

Example 19

medium
A point lies on the line y=xy = x and is 5 units from the origin. Find its possible positions.

Example 20

hard
A triangle has sides aa, bb, cc with perimeter 3030. Write all the geometric constraints (triangle inequality and perimeter) that aa, bb, cc must satisfy, then find the maximum possible value of side aa given b=cb = c.
← Back to Geometric Constraints Worked Examples