Coordinate Geometry Proofs: Geometry Math Test – PDF Download

Question 1: Distance and midpoint (2 points)

Points \(A(-4,3)\) and \(B(2,-5)\) are given.

1. Find the distance \(AB\). (1 point)
2. Find the midpoint of \(\overline{AB}\). (1 point)

Question 2: Parallel or perpendicular (2 points)

Line 1 passes through \((1,-2)\) and \((4,4)\). Line 2 passes through \((-1,5)\) and \((3,3)\). Line 3 passes through \((0,0)\) and \((2,4)\). Identify every pair of these lines that is parallel and every pair that is perpendicular. Show the slopes.

Question 3: Perpendicular line (2 points)

Write an equation of the line through \((3,5)\) that is perpendicular to the line \(y=-\frac{3}{4}x+2\).

Question 4: Partition a segment (2 points)

Segment \(\overline{AB}\) has endpoints \(A(-7,4)\) and \(B(5,-2)\).

1. Find the point that partitions \(\overline{AB}\) in the ratio \(1:2\) from \(A\). (1 point)
2. Find the point that partitions \(\overline{AB}\) in the ratio \(3:1\) from \(A\). (1 point)

Question 5: Triangle perimeter and area (3 points)

Triangle \(PQR\) has vertices \(P(-2,-1)\), \(Q(6,-1)\) and \(R(2,5)\).

1. Find the perimeter. Give the exact value and the value rounded to the nearest hundredth. (2 points)
2. Find the area. (1 point)

Question 6: Missing endpoint (2 points)

\(M(1,-2)\) is the midpoint of \(\overline{AB}\) and \(A\) is the point \((-5,4)\).

1. Find the coordinates of \(B\). (1 point)
2. Find the exact length of \(\overline{AB}\). (1 point)

Question 7: Prove a rectangle (3 points)

Quadrilateral \(JKLM\) has vertices \(J(-3,0)\), \(K(1,-2)\), \(L(4,4)\) and \(M(0,6)\). Prove that \(JKLM\) is a rectangle, then find its area.

Question 8: Campus map (2 points)

On Priya’s school map each unit is 50 feet. The gym is at \(G(1,2)\) and the cafeteria is at \(C(9,8)\). A straight walkway joins them.

1. How long is the walkway in feet? (1 point)
2. A bench is placed \(\frac{3}{4}\) of the way from the gym to the cafeteria. Find the coordinates of the bench. (1 point)

Question 9: Classify a triangle (2 points)

Triangle \(ABC\) has vertices \(A(-2,2)\), \(B(1,6)\) and \(C(5,3)\). Classify it as scalene, isosceles or equilateral, and decide whether it is a right triangle. Justify both answers.

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