Geometry Practice Test: Coordinate Geometry Proofs
- Coordinate Geometry Proofs
- 10 questions
- 15 minutes
Each question uses the slope, distance or midpoint formula to prove something about a figure. Tip: equal slopes show parallel sides, slopes with a product of −1 show perpendicular sides, and equal distances show congruent sides.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
In a coordinate proof, which result is enough to show that quadrilateral ABCD is a parallelogram?
Correct answer: B
Equal slopes on both pairs of opposite sides mean both pairs are parallel, which is the definition of a parallelogram. Equal diagonals or perpendicular diagonals alone can also happen in an isosceles trapezoid or a kite.
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Question 2
Triangle ABC has vertices A(1, 1), B(4, 5) and C(8, 2).
Which is the most specific correct classification?
Correct answer: C
AB = 9 + 16 = 5 and BC = 16 + 9 = 5, so the triangle is isosceles. The slopes of AB (4/3) and BC (−3/4) multiply to −1, so ∠B is a right angle.
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Question 3
Quadrilateral PQRS has vertices P(1, 2), Q(4, 6), R(9, 6) and S(6, 2).
Which is the most specific correct name for PQRS?
Correct answer: D
All four sides are 5 units long (for example, PQ = 9 + 16 = 5 and QR = 5), so it is a rhombus. The slope of PQ is 4/3 and QR is horizontal, so the angle at Q is not a right angle and it is not a square.
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Question 4
A parallelogram is placed with vertices A(0, 0), B(a, 0), C(a + b, c) and D(b, c). To prove that the diagonals bisect each other, you find the midpoint of each diagonal.
What is the midpoint of diagonal AC?
Correct answer: A
The midpoint averages the coordinates of (0, 0) and (a + b, c), giving ((a + b)/2, c/2). The midpoint of BD is the same point, which proves that the diagonals bisect each other.
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Question 5
A right triangle has vertices O(0, 0), A(a, 0) and B(0, b). M is the midpoint of hypotenuse AB.
What is the distance from M to O?
Correct answer: C
M = (a/2, b/2), so MO = a²/4 + b²/4 = ½a² + b². This is half the hypotenuse, which proves the midpoint of the hypotenuse is equally far from all three vertices.
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Question 6
Segment AB has endpoints A(−1, 4) and B(3, −2).
Which line is perpendicular to AB?
Correct answer: D
The slope of AB is (−2 − 4)/(3 − (−1)) = −6/4 = −3/2. A perpendicular line has the negative reciprocal slope, 2/3. A slope of −3/2 would be parallel to AB.
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Question 7
Quadrilateral ABCD has vertices A(0, 0), B(8, 0), C(6, 4) and D(2, 4).
Which is the most specific correct name for ABCD?
Correct answer: B
AB and DC are both horizontal, so they are parallel, while AD (slope 2) and BC (slope −2) are not parallel: it is a trapezoid. The legs are equal, AD = BC = 20, so it is isosceles.
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Question 8
Points A(1, 2), B(5, 3) and C(7, 7) are three vertices of parallelogram ABCD, in that order.
What are the coordinates of D?
Correct answer: A
In parallelogram ABCD, the move from B to A equals the move from C to D: (−4, −1). So D = (7 − 4, 7 − 1) = (3, 6). The point (11, 8) would make ABDC, a different vertex order.
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Question 9
Points A(2, 1) and B(6, 5) are given.
Which point is the same distance from A as from B?
Correct answer: C
For (0, 7): PA² = 4 + 36 = 40 and PB² = 36 + 4 = 40, so the distances are equal. It lies on the perpendicular bisector of AB, y = −x + 7. Each of the other points is closer to A.
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Question 10
Triangle OAB has vertices O(0, 0), A(2a, 0) and B(2b, 2c). M and N are the midpoints of OB and AB, so M(b, c) and N(a + b, c).
What does this placement prove about segment MN?
Correct answer: D
MN has slope 0, like OA, so the segments are parallel. Its length is (a + b) − b = a, half of OA = 2a. This is the Triangle Midsegment Theorem.
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