These 25 problems practice coordinate geometry for Geometry students, aligned to G-GPE.B.4, G-GPE.B.5, G-GPE.B.6 and G-GPE.B.7. They begin with the distance and midpoint formulas and move through slopes and equations of parallel and perpendicular lines.
Later problems partition segments in a ratio, find perimeter and area, and prove that triangles and quadrilaterals have a given type. The last six are word problems and multi-step tasks.
Problem 1: Vertical distance
Find the length of the segment with endpoints \((-3,4)\) and \((-3,-6)\).
Problem 2: Distance with a 3-4-5 triangle
Use the distance formula to find the distance between \((1,2)\) and \((4,6)\).
Problem 3: Distance between two points
Find the distance between \((-2,-1)\) and \((4,7)\).
Problem 4: Distance as a radical
Find the exact distance between \((-3,2)\) and \((1,4)\), then round to the nearest hundredth.
Problem 5: Midpoint of a segment
Find the midpoint of the segment with endpoints \((2,8)\) and \((10,-4)\).
Problem 6: Midpoint with fractions
Find the midpoint of the segment with endpoints \((-5,3)\) and \((2,6)\).
Problem 7: Missing endpoint
The midpoint of \(\overline{AB}\) is \(M(3,-1)\) and the endpoint \(A\) is \((-2,4)\). Find the coordinates of \(B\).
Problem 8: Slopes of two lines
Line \(a\) passes through \((1,2)\) and \((4,8)\). Line \(b\) passes through \((0,5)\) and \((6,2)\). Decide whether the lines are parallel, perpendicular, or neither.
Problem 9: Parallel line through a point
Write an equation of the line through \((4,-1)\) that is parallel to \(y=3x-2\).
Problem 10: Perpendicular line through a point
Write an equation of the line through \((6,2)\) that is perpendicular to \(y=-\frac{2}{3}x+1\).
Problem 11: Point one third of the way
Find the point \(P\) on \(\overline{AB}\) that partitions the segment from \(A(0,2)\) to \(B(12,14)\) in the ratio \(1:2\).
Problem 12: Ratio 3 to 2
Find the point that partitions the segment from \(A(-6,2)\) to \(B(4,-8)\) in the ratio \(3:2\) measured from \(A\).
Problem 13: Perimeter of a triangle
Triangle \(ABC\) has vertices \(A(0,0)\), \(B(6,0)\) and \(C(6,8)\). Find its perimeter.
Problem 14: Area of a tilted rectangle
Rectangle \(PQRS\) has vertices \(P(0,0)\), \(Q(4,2)\), \(R(3,4)\) and \(S(-1,2)\). Find its area.
Problem 15: Area of a triangle
Find the area of the triangle with vertices \(A(-2,1)\), \(B(6,1)\) and \(C(1,7)\).
Problem 16: Classify a triangle by sides
Triangle \(ABC\) has vertices \(A(1,1)\), \(B(7,1)\) and \(C(4,6)\). Use the distance formula to decide whether it is scalene, isosceles, or equilateral.
Problem 17: Right angle with slopes
Triangle \(PQR\) has vertices \(P(1,1)\), \(Q(5,3)\) and \(R(3,7)\). Show that it is a right triangle and find its area.
Problem 18: Prove a parallelogram
Quadrilateral \(ABCD\) has vertices \(A(1,1)\), \(B(6,2)\), \(C(8,6)\) and \(D(3,5)\). Prove that it is a parallelogram by finding the midpoints of its diagonals.
Problem 19: Prove a rhombus
Prove that the quadrilateral with vertices \(A(-3,0)\), \(B(0,4)\), \(C(3,0)\) and \(D(0,-4)\) is a rhombus, then find its area.
Problem 20: Walk to the library
On a map of Maya’s town, each unit is one block and each block is 0.1 mile. The school is at \(S(2,3)\) and the library is at \(L(14,8)\). (a) Find the straight-line distance from the school to the library in miles. (b) A bench sits halfway between them. Find its coordinates.
Problem 21: Ranger station on a trail
A straight trail runs from the trailhead \(T(0,0)\) to a campsite \(C(15,20)\) on a map where each unit is 0.1 mile. The ranger station \(R\) is on the trail, \(\frac{2}{5}\) of the way from the trailhead to the campsite. Find the coordinates of \(R\) and the distance from \(R\) to the campsite in miles.
Problem 22: Garden fence cost
Diego plans a rectangular garden with corners \(A(2,1)\), \(B(10,1)\), \(C(10,7)\) and \(D(2,7)\) on a plan where each unit is 1 foot. (a) Find the perimeter. (b) Fencing costs $4.50 per foot; find the total cost. (c) Find the area. (d) A path runs from \(A\) to \(C\); find its length.
Problem 23: Isosceles trapezoid park
A park has corners \(A(0,0)\), \(B(8,0)\), \(C(6,4)\) and \(D(2,4)\) on a map where each unit is 10 yards. Prove that \(ABCD\) is an isosceles trapezoid, then find its area in square yards.
Problem 24: Split a triangle
Triangle \(ABC\) has vertices \(A(0,0)\), \(B(12,0)\) and \(C(4,9)\). Point \(D\) partitions \(\overline{AB}\) in the ratio \(1:2\) from \(A\). (a) Find \(D\). (b) Show that \(\overline{CD}\) is perpendicular to \(\overline{AB}\). (c) Find the area of triangle \(ABC\). (d) Find the perimeter of triangle \(ACD\) to the nearest hundredth.
Problem 25: Drone flight
Jamal’s drone takes off from \(P(-4,-2)\) and flies in a straight line to a landing spot \(L(8,14)\) on a map where each unit is 10 yards. (a) How many yards does it fly? (b) It pauses to take a photo at the midpoint of the flight. Find the coordinates of that point.
See every chapter on the Geometry math page.