Math 1 Quiz: Midpoint Formula
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Midpoint FormulaQuestion 1 of 8

A city planning committee is designing a new park layout on a coordinate grid where each unit represents 100 meters. The park will have three main features: a playground, a pond, and a pavilion.

The playground is located at (−2,5)(-2, 5) and the pavilion is at (4,−1)(4, -1). A circular pond will be constructed so that its center is equidistant from both the playground and pavilion. If the pond's center must also be located at the point where x and y coordinates are both integers, and it lies on the perpendicular bisector of the segment connecting the playground and pavilion, what are the coordinates of the pond's center?

(1,2)(1, 2)
(2,1)(2, 1)
(0,3)(0, 3)
(3,0)(3, 0)
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Math 1 Quiz

Math 1 Quiz: Midpoint Formula

Practice Midpoint Formula in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Midpoint Formula, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A city planning committee is designing a new park layout on a coordinate grid where each unit represents 100 meters. The park will have three main features: a playground, a pond, and a pavilion.

The playground is located at (−2,5)(-2, 5) and the pavilion is at (4,−1)(4, -1). A circular pond will be constructed so that its center is equidistant from both the playground and pavilion. If the pond's center must also be located at the point where x and y coordinates are both integers, and it lies on the perpendicular bisector of the segment connecting the playground and pavilion, what are the coordinates of the pond's center?

  1. (1,2)(1, 2) (correct answer)
  2. (2,1)(2, 1)
  3. (0,3)(0, 3)
  4. (3,0)(3, 0)
Explanation: The center of the pond lies on the perpendicular bisector of the segment from (−2,5)(-2, 5) to (4,−1)(4, -1). First, find the midpoint: (−2+42,5+(−1)2)=(1,2)\left(\frac{-2+4}{2}, \frac{5+(-1)}{2}\right) = (1, 2). The slope of the segment is −1−54−(−2)=−66=−1\frac{-1-5}{4-(-2)} = \frac{-6}{6} = -1. The perpendicular bisector has slope 11 (negative reciprocal) and passes through (1,2)(1, 2). Its equation is y−2=1(x−1)y - 2 = 1(x - 1), or y=x+1y = x + 1. We need integer coordinates on this line. The point (1,2)(1, 2) itself satisfies y=x+1y = x + 1 since 2=1+12 = 1 + 1. Let's verify it's equidistant from both points: distance to (−2,5)(-2, 5) is (1−(−2))2+(2−5)2=9+9=32\sqrt{(1-(-2))^2 + (2-5)^2} = \sqrt{9+9} = 3\sqrt{2}. Distance to (4,−1)(4, -1) is (1−4)2+(2−(−1))2=9+9=32\sqrt{(1-4)^2 + (2-(-1))^2} = \sqrt{9+9} = 3\sqrt{2}. Since both distances equal 323\sqrt{2}, the pond center is at (1,2)(1, 2). The other choices don't lie on the perpendicular bisector or aren't equidistant from both points.

Question 2

A surveyor is mapping a triangular plot of land with vertices at P(−2,3)(-2, 3), Q(4,7)(4, 7), and R(6,−1)(6, -1). She needs to place a marker at the point that is equidistant from the midpoints of sides PQ and QR. If this marker is at coordinates (a,b)(a, b), what is the value of a+ba + b?

  1. 8.58.5
  2. 66
  3. 77 (correct answer)
  4. 5.55.5
Explanation: First, find the midpoint of PQ: MPQ=(−2+42,3+72)=(1,5)M_{PQ} = \left(\frac{-2+4}{2}, \frac{3+7}{2}\right) = (1, 5). Next, find the midpoint of QR: MQR=(4+62,7+(−1)2)=(5,3)M_{QR} = \left(\frac{4+6}{2}, \frac{7+(-1)}{2}\right) = (5, 3). The point equidistant from these two midpoints is the midpoint of the segment connecting them: (1+52,5+32)=(3,4)\left(\frac{1+5}{2}, \frac{5+3}{2}\right) = (3, 4). Therefore, a=3a = 3 and b=4b = 4, so a+b=7a + b = 7. Choice A results from incorrectly averaging all original coordinates. Choice B comes from adding coordinates of one midpoint. Choice D represents the average of the x-coordinates of the two midpoints only.

Question 3

In a coordinate plane, points A, B, and C are collinear with B between A and C. If A is at (−3,2)(-3, 2), C is at (9,−4)(9, -4), and the distance from A to B is twice the distance from B to C, what are the coordinates of point B?

  1. (3,−2)(3, -2)
  2. (1,0)(1, 0)
  3. (5,−2)(5, -2) (correct answer)
  4. (7,−10/3)(7, -10/3)
Explanation: Since B divides AC in the ratio 2:1 (distance AB is twice distance BC), B divides AC internally in the ratio 2:1. Using the section formula: B=(1⋅xA+2⋅xC1+2,1⋅yA+2⋅yC1+2)=(−3+2(9)3,2+2(−4)3)=(153,−63)=(5,−2)B = \left(\frac{1 \cdot x_A + 2 \cdot x_C}{1 + 2}, \frac{1 \cdot y_A + 2 \cdot y_C}{1 + 2}\right) = \left(\frac{-3 + 2(9)}{3}, \frac{2 + 2(-4)}{3}\right) = \left(\frac{15}{3}, \frac{-6}{3}\right) = (5, -2). Choice A gives the midpoint of AC. Choice B results from incorrectly using equal division. Choice D comes from using the wrong ratio or calculation error.

Question 4

Triangle ABC has vertices A(0,0)(0, 0), B(6,8)(6, 8), and C(12,0)(12, 0). Point M is the midpoint of side AB, and point N is the midpoint of side BC. If a line segment connects M to N, what is the length of segment MN?

  1. 33
  2. 66 (correct answer)
  3. 44
  4. 55
Explanation: First, find the coordinates of M (midpoint of AB): M=(0+62,0+82)=(3,4)M = \left(\frac{0+6}{2}, \frac{0+8}{2}\right) = (3, 4). Next, find N (midpoint of BC): N=(6+122,8+02)=(9,4)N = \left(\frac{6+12}{2}, \frac{8+0}{2}\right) = (9, 4). The length of MN is: (9−3)2+(4−4)2=36+0=6\sqrt{(9-3)^2 + (4-4)^2} = \sqrt{36 + 0} = 6. Note that MN is parallel to AC (both horizontal) and has length equal to half the length of AC, which is consistent with the midpoint theorem. AC has length 12−0=1212 - 0 = 12, so MN should be 122=6\frac{12}{2} = 6. Choice A represents half the correct answer. Choice C is the y-coordinate of both midpoints. Choice D might result from calculation errors.

Question 5

A landscaper places three sprinklers at points A(4,12)A(4, 12), B(16,8)B(16, 8), and C(−8,4)C(-8, 4). If a fourth sprinkler is placed at point DD such that the midpoint of segment ACAC is the same as the midpoint of segment BDBD, what are the coordinates of point DD?

  1. (−20,8)(-20, 8) (correct answer)
  2. (−4,12)(-4, 12)
  3. (12,4)(12, 4)
  4. (−12,12)(-12, 12)
Explanation: First, find the midpoint of segment AC: MAC=(4+(−8)2,12+42)=(−42,162)=(−2,8)M_{AC} = \left(\frac{4+(-8)}{2}, \frac{12+4}{2}\right) = \left(\frac{-4}{2}, \frac{16}{2}\right) = (-2, 8). Since the midpoint of BD must equal the midpoint of AC, we have: (16+xD2,8+yD2)=(−2,8)\left(\frac{16+x_D}{2}, \frac{8+y_D}{2}\right) = (-2, 8). This gives us two equations: 16+xD2=−2\frac{16+x_D}{2} = -2 and 8+yD2=8\frac{8+y_D}{2} = 8. Solving the first equation: 16+xD=−416+x_D = -4, so xD=−20x_D = -20. Solving the second equation: 8+yD=168+y_D = 16, so yD=8y_D = 8. Therefore, D=(−20,8)D = (-20, 8). Choice B would result from incorrectly using the coordinates of A. Choice C would result from incorrectly swapping coordinates. Choice D would result from sign errors in the calculation.

Question 6

On a map, City Hall is located at (6,15)(6, 15) and the Fire Station is at (18,3)(18, 3). A new Police Station will be built at the midpoint between these two buildings. If each unit on the map represents 0.5 miles, what is the actual distance in miles from the Police Station to City Hall?

  1. 3.03.0 miles
  2. 4.54.5 miles (correct answer)
  3. 6.06.0 miles
  4. 7.57.5 miles
Explanation: First, find the midpoint (Police Station location): P=(6+182,15+32)=(242,182)=(12,9)P = \left(\frac{6+18}{2}, \frac{15+3}{2}\right) = \left(\frac{24}{2}, \frac{18}{2}\right) = (12, 9). Next, find the distance from the Police Station at (12,9)(12, 9) to City Hall at (6,15)(6, 15) in map units: d=(12−6)2+(9−15)2=62+(−6)2=36+36=72=62≈8.49d = \sqrt{(12-6)^2 + (9-15)^2} = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} ≈ 8.49 units. Since each unit represents 0.5 miles, the actual distance is: 62×0.5=32≈4.246\sqrt{2} \times 0.5 = 3\sqrt{2} ≈ 4.24 miles. Since 2≈1.414\sqrt{2} ≈ 1.414, we have 32≈3(1.414)=4.2423\sqrt{2} ≈ 3(1.414) = 4.242 miles, which rounds to 4.5 miles. Choice A represents half the correct distance. Choice C would result from forgetting to apply the scale factor. Choice D would result from adding instead of using the distance formula.

Question 7

Points P and Q are endpoints of a diameter of a circle. If P is at (−5,2)(-5, 2) and the center of the circle is at (1,−3)(1, -3), what are the coordinates of point Q?

  1. (7,−8)(7, -8) (correct answer)
  2. (3,−4)(3, -4)
  3. (−2,−0.5)(-2, -0.5)
  4. (6,−1)(6, -1)
Explanation: Since P and Q are endpoints of a diameter, the center of the circle is the midpoint of segment PQ. Using the midpoint formula: center (1,−3)=(−5+xQ2,2+yQ2)(1, -3) = \left(\frac{-5 + x_Q}{2}, \frac{2 + y_Q}{2}\right). Solving for Q: 1=−5+xQ21 = \frac{-5 + x_Q}{2} gives xQ=7x_Q = 7, and −3=2+yQ2-3 = \frac{2 + y_Q}{2} gives yQ=−8y_Q = -8. Therefore Q is at (7,−8)(7, -8). Choice B gives the coordinates if you average P and the center. Choice C results from incorrectly halving the center coordinates. Choice D comes from adding rather than using the midpoint relationship correctly.

Question 8

In triangle PQR, the midpoint of side PQ is M(2,3)(2, 3), the midpoint of side QR is N(5,1)(5, 1), and the midpoint of side PR is L(0,−1)(0, -1). What is the area of triangle PQR?

  1. 2020
  2. 2424 (correct answer)
  3. 1616
  4. 2828
Explanation: The triangle formed by connecting the midpoints of a triangle has area equal to 14\frac{1}{4} the area of the original triangle. First, find the area of triangle MLN using the coordinate formula: Area=12∣xM(yL−yN)+xL(yN−yM)+xN(yM−yL)∣\text{Area} = \frac{1}{2}|x_M(y_L - y_N) + x_L(y_N - y_M) + x_N(y_M - y_L)|. Substituting M(2,3)(2, 3), L(0,−1)(0, -1), N(5,1)(5, 1): AreaMLN=12∣2(−1−1)+0(1−3)+5(3−(−1))∣=12∣2(−2)+0(−2)+5(4)∣=12∣−4+0+20∣=12⋅16=6\text{Area}_{MLN} = \frac{1}{2}|2(-1-1) + 0(1-3) + 5(3-(-1))| = \frac{1}{2}|2(-2) + 0(-2) + 5(4)| = \frac{1}{2}|-4 + 0 + 20| = \frac{1}{2} \cdot 16 = 6. Since the area of triangle MLN is 14\frac{1}{4} the area of triangle PQR, we have: 6=14⋅AreaPQR6 = \frac{1}{4} \cdot \text{Area}_{PQR}, so AreaPQR=24\text{Area}_{PQR} = 24. The other choices represent common calculation errors or incorrect ratios between the areas.