Geometry Quiz: Coordinates To Find Perimeter And Area
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Coordinates To Find Perimeter And AreaQuestion 1 of 20

Rectangle WXYZ has W(2, 1), X(2, 7), Y(8, 7), and Z(8, 1). If point P is at (5, 4), what is the area of the region inside rectangle WXYZ but outside triangle WPZ?

27 square units
30 square units
33 square units
36 square units
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Geometry Quiz

Geometry Quiz: Coordinates To Find Perimeter And Area

Practice Coordinates To Find Perimeter And Area in Geometry 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 Coordinates To Find Perimeter And Area, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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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

Rectangle WXYZ has W(2, 1), X(2, 7), Y(8, 7), and Z(8, 1). If point P is at (5, 4), what is the area of the region inside rectangle WXYZ but outside triangle WPZ?

  1. 27 square units (correct answer)
  2. 30 square units
  3. 33 square units
  4. 36 square units
Explanation: The correct answer is A. First find the area of rectangle WXYZ: length = 8-2 = 6, width = 7-1 = 6, so area = 36 square units. Next find the area of triangle WPZ with vertices W(2,1), P(5,4), Z(8,1). Using the triangle area formula: Area = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| = ½|2(4-1) + 5(1-1) + 8(1-4)| = ½|6 + 0 - 24| = ½|18| = 9 square units. The area inside the rectangle but outside the triangle is 36 - 9 = 27 square units. Choice B results from incorrectly calculating the triangle area as 6. Choice C results from calculating the triangle area as 3. Choice D gives the full rectangle area without subtracting the triangle.

Question 2

Triangle ABC has vertices at A(2, 3), B(8, 7), and C(5, 11). If the triangle is reflected across the y-axis to form triangle A'B'C', what is the difference between the perimeter of triangle ABC and the perimeter of triangle A'B'C'?

  1. 0 (correct answer)
  2. 4
  3. 8
  4. 12
Explanation: The correct answer is A. Reflections preserve distances, so the perimeter remains unchanged. Using the distance formula: AB = √[(8-2)² + (7-3)²] = √52, BC = √[(5-8)² + (11-7)²] = √25 = 5, and AC = √[(5-2)² + (11-3)²] = √73. After reflection across the y-axis, A'(-2,3), B'(-8,7), C'(-5,11) form a congruent triangle with identical side lengths, so the difference is 0. Choice B incorrectly uses the difference in x-coordinates of one vertex. Choice C doubles this error. Choice D represents three times the coordinate difference, possibly from adding coordinate changes.

Question 3

Quadrilateral ABCDABCD has vertices A(0,0)A(0,0), B(4,0)B(4,0), C(3,3)C(3,3), and D(−1,3)D(-1,3) in that order. What is the perimeter of the figure?

  1. 1414
  2. 4+3+4+34+3+4+3
  3. 4+10+4+104+\sqrt{10}+4+\sqrt{10} (correct answer)
  4. 4+18+4+184+\sqrt{18}+4+\sqrt{18}
Explanation: The skill is finding the perimeter of a quadrilateral using coordinates. The vertices are A(0,0), B(4,0), C(3,3), and D(-1,3). Side lengths are computed using the distance formula, which is the square root of the sum of the squares of the differences in x and y coordinates. The perimeter is the sum of all four side lengths. Calculating each distance gives AB = 4, BC = √10, CD = 4, and DA = √10, justifying the final value as 4 + √10 + 4 + √10. A common distractor misconception is using √18 instead of √10 by doubling a difference. To transfer this strategy, compute side lengths before summing for perimeter or using them for area in other figures.

Question 4

A right triangle has vertices at O(0, 0), P(6, 0), and Q(0, 8). If this triangle is rotated 90° counterclockwise about the origin to form triangle O'P'Q', what is the area of the quadrilateral formed by the union of both triangles?

  1. 24 square units
  2. 36 square units
  3. 48 square units (correct answer)
  4. 60 square units
Explanation: The correct answer is C. The original triangle OPQ has area ½(6)(8) = 24. After 90° counterclockwise rotation about origin: O'(0,0), P'(0,6), Q'(-8,0). The rotated triangle O'P'Q' also has area 24. However, the triangles overlap only at the origin, so the total area of their union is 24 + 24 = 48. Choice A gives the area of just one triangle. Choice B incorrectly assumes some overlap beyond the origin point. Choice D might result from adding the areas and including additional area incorrectly.

Question 5

Points A(−1,2)A(-1,2), B(3,1)B(3,1), and C(1,5)C(1,5) are plotted on a coordinate plane and connected to form triangle ABCABC (segments AB‾\overline{AB}, BC‾\overline{BC}, and CA‾\overline{CA}). What is the perimeter of the figure?

  1. 4+17+134+\sqrt{17}+\sqrt{13}
  2. 17+20+13\sqrt{17}+\sqrt{20}+\sqrt{13} (correct answer)
  3. 17+20\sqrt{17}+\sqrt{20}
  4. 17+20+1317+20+13
Explanation: The skill is finding the perimeter of a triangle using coordinates. The vertices are A(-1,2), B(3,1), and C(1,5). Side lengths are computed using the distance formula, which is the square root of the sum of the squares of the differences in x and y coordinates. The perimeter is the sum of all three side lengths. Calculating each distance gives AB = √17, BC = √20, and CA = √13, justifying the final value as √17 + √20 + √13. A common distractor misconception is adding the squared distances without taking square roots, resulting in 17 + 20 + 13. To transfer this strategy, compute side lengths before summing for perimeter or using them for area in other figures.

Question 6

Triangle ABC has vertices A(1, 2), B(7, 5), and C(4, 8). The triangle is translated 3 units left and 2 units down to form triangle A'B'C'. What is the ratio of the area of triangle ABC to the area of triangle A'B'C'?

  1. 9:4
  2. 3:2
  3. 5:1
  4. 1:1 (correct answer)
Explanation: When you encounter transformation problems in geometry, the key insight is understanding how different transformations affect area and other measurements. Translation is a transformation that slides a figure from one position to another without changing its size, shape, or orientation. When triangle ABC is translated 3 units left and 2 units down, every point moves the same distance in the same direction. The new vertices A'B'C' are simply the original vertices with each x-coordinate decreased by 3 and each y-coordinate decreased by 2. Since translation preserves all distances and angles, the translated triangle A'B'C' is congruent to the original triangle ABC. Congruent figures have identical areas, so the ratio of their areas is 1:11:1. Looking at the incorrect choices: Answer A) 9:49:4 might tempt you if you incorrectly squared the translation distances (3² and 2²), but translation distances don't affect area ratios. Answer B) 3:23:2 could mislead you into thinking the ratio relates directly to the translation amounts (3 units and 2 units), but again, translation preserves area. Answer C) 5:15:1 has no logical connection to this problem and represents a common distractor with no mathematical basis. Remember this crucial distinction: translations, rotations, and reflections (rigid transformations) preserve area, while dilations (scaling transformations) change area by the square of the scale factor. When you see translation problems, immediately think "same area, same shape" – the ratio will always be 1:11:1.

Question 7

Quadrilateral PQRSPQRS has vertices P(1,2)P(1,2), Q(6,2)Q(6,2), R(6,7)R(6,7), and S(1,7)S(1,7) in the coordinate plane (units are uniform). What is the perimeter of the figure?

  1. 2525
  2. 2020 (correct answer)
  3. 5050
  4. 1010
Explanation: This problem asks for the perimeter of quadrilateral PQRS with vertices P(1,2), Q(6,2), R(6,7), and S(1,7). The vertices are P(1,2), Q(6,2), R(6,7), and S(1,7), forming a rectangle. Since P and Q have the same y-coordinate (y=2), PQ is horizontal with length |6-1| = 5. Since Q and R have the same x-coordinate (x=6), QR is vertical with length |7-2| = 5. Similarly, RS is horizontal with length |6-1| = 5, and SP is vertical with length |7-2| = 5. The perimeter is 5 + 5 + 5 + 5 = 20. A common error is calculating the area (5×5 = 25) instead of perimeter. The key insight is recognizing this is a square with side length 5.

Question 8

Refer to the coordinate plane below. Pentagon ABCDEABCDE has vertices A(0,0)A(0,0), B(6,0)B(6,0), C(6,4)C(6,4), D(3,7)D(3,7), and E(0,4)E(0,4). What is the area of the pentagon?

  1. 2727 square units
  2. 3333 square units (correct answer)
  3. 3636 square units
  4. 4242 square units
Explanation: Decompose into rectangle ABCEABCE with area 6×4=246 \times 4 = 24 and triangle CDECDE with base CE=6CE = 6 and height =7−4=3= 7-4 = 3, area =12(6)(3)=9= \frac{1}{2}(6)(3) = 9. Total =33= 33. Choice A omits the rectangle's full width. Choice C forgets to halve the triangle. Choice D adds an extra unit strip.

Question 9

Refer to the figure below. Quadrilateral JKLMJKLM has vertices J(−4,−1)J(-4, -1), K(−1,3)K(-1, 3), L(4,3)L(4, 3), and M(1,−1)M(1, -1). What is the perimeter of JKLMJKLM?

  1. 1616 units
  2. 1818 units
  3. 2020 units (correct answer)
  4. 2525 units
Explanation: JK=9+16=5JK = \sqrt{9+16} = 5; KL=5KL = 5 (horizontal); LM=9+16=5LM = \sqrt{9+16} = 5; MJ=5MJ = 5 (horizontal). Perimeter =20= 20. Choice A adds only two pairs incorrectly. Choice B miscomputes one diagonal side as 4. Choice D uses the sum of squared distances instead.

Question 10

Based on the coordinate plane shown, what is the area of triangle DEF?

  1. 12 square units
  2. 15 square units (correct answer)
  3. 18 square units
  4. 21 square units
Explanation: The correct answer is B. Using the coordinate formula for triangle area with D(-2, 1), E(4, 5), F(2, -3): Area = ¬Ω|x‚ÇÅ(y‚ÇÇ-y‚ÇÉ) + x‚ÇÇ(y‚ÇÉ-y‚ÇÅ) + x‚ÇÉ(y‚ÇÅ-y‚ÇÇ)| = ¬Ω|(-2)(5-(-3)) + 4(-3-1) + 2(1-5)| = ¬Ω|(-2)(8) + 4(-4) + 2(-4)| = ¬Ω|-16 - 16 - 8| = ¬Ω|-30| = 15. Choice A results from using base√óheight incorrectly with a horizontal distance of 6 and estimated height of 4. Choice C comes from calculating the area of the bounding rectangle minus triangular corners incorrectly. Choice D represents a common error in the shoelace formula by forgetting the ¬Ω factor.

Question 11

Rectangle PQRS has vertices P(1, 2), Q(7, 2), R(7, 6), and S(1, 6). Point T is located at (4, 9). What is the area of quadrilateral PQTS?

  1. 15 square units
  2. 18 square units (correct answer)
  3. 21 square units
  4. 24 square units
Explanation: The correct answer is B. Using the shoelace formula for quadrilateral PQTS with vertices P(1,2), Q(7,2), T(4,9), S(1,6): Area = ½|x₁(y₂-y₄) + x₂(y₃-y₁) + x₃(y₄-y₂) + x₄(y₁-y₃)| = ½|1(2-6) + 7(9-2) + 4(6-2) + 1(2-9)| = ½|(-4) + 49 + 16 + (-7)| = ½|36| = 18. Choice A results from incorrectly calculating the rectangle area as 3×5. Choice C comes from adding the rectangle area (24) and subtracting 3 incorrectly. Choice D is simply the area of rectangle PQRS, ignoring point T.

Question 12

Quadrilateral ABCD has vertices A(0,1)A(0,1), B(4,4)B(4,4), C(7,0)C(7,0), and D(3,−3)D(3,-3) connected in that order. What is the perimeter of the figure?

  1. 5+5+5+5=205+5+5+5=20
  2. (4−0)2+(4−1)2+(7−4)2+(0−4)2+(3−7)2+(−3−0)2+(0−3)2+(1−(−3))2\sqrt{(4-0)^2+(4-1)^2}+\sqrt{(7-4)^2+(0-4)^2}+\sqrt{(3-7)^2+(-3-0)^2}+\sqrt{(0-3)^2+(1-(-3))^2} (correct answer)
  3. (4−0)2+(4−1)2+(7−4)2+(0−4)2+(3−7)2+(−3−0)2\sqrt{(4-0)^2+(4-1)^2}+\sqrt{(7-4)^2+(0-4)^2}+\sqrt{(3-7)^2+(-3-0)^2}
  4. 12∣0(4−0)+4(0+3)+7(−3−1)+3(1−4)∣\frac{1}{2}\left|0(4-0)+4(0+3)+7(-3-1)+3(1-4)\right|
Explanation: The skill is finding the perimeter of a quadrilateral using coordinates. The vertices are A(0,1)A(0,1), B(4,4)B(4,4), C(7,0)C(7,0), and D(3,−3)D(3,-3). Side lengths are computed using the distance formula between consecutive points. The perimeter is the sum of these four side lengths. Each simplifies to 5, summing to 20, justifying the expression in the correct choice. A distractor misconception is summing only three sides, as in choice C. To transfer this strategy, always compute individual side lengths before summing for the perimeter.

Question 13

A rectangle RSTU is shown with vertices R(−2,−3)R(-2,-3), S(4,−3)S(4,-3), T(4,1)T(4,1), and U(−2,1)U(-2,1). Which calculation is required to find the area?

  1. (4−(−2))+(1−(−3))(4-(-2))+(1-(-3))
  2. 2((4−(−2))+(1−(−3)))2\big((4-(-2))+(1-(-3))\big)
  3. (4−(−2))⋅(1−(−3))(4-(-2))\cdot(1-(-3)) (correct answer)
  4. (4−(−2))2+(1−(−3))2\sqrt{(4-(-2))^2+(1-(-3))^2}
Explanation: The skill is finding the area of a rectangle using coordinates. The vertices are R(-2,-3), S(4,-3), T(4,1), and U(-2,1). Side lengths are computed by differences in x and y for length and width. The area logic is length times width. This gives 6 times 4 equals 24, justifying the calculation in the correct choice. A distractor misconception is confusing area with perimeter, as in choice B. To transfer this strategy, compute length and width before multiplying for area.

Question 14

Quadrilateral ABCD has vertices A(0,1)A(0,1), B(4,4)B(4,4), C(7,0)C(7,0), and D(3,−3)D(3,-3) connected in that order. What is the perimeter of the figure?​​

  1. 5+5+5+5=205+5+5+5=20
  2. (4−0)2+(4−1)2+(7−4)2+(0−4)2+(3−7)2+(−3−0)2+(0−3)2+(1−(−3))2\sqrt{(4-0)^2+(4-1)^2}+\sqrt{(7-4)^2+(0-4)^2}+\sqrt{(3-7)^2+(-3-0)^2}+\sqrt{(0-3)^2+(1-(-3))^2} (correct answer)
  3. (4−0)2+(4−1)2+(7−4)2+(0−4)2+(3−7)2+(−3−0)2\sqrt{(4-0)^2+(4-1)^2}+\sqrt{(7-4)^2+(0-4)^2}+\sqrt{(3-7)^2+(-3-0)^2}
  4. 12∣0(4−0)+4(0+3)+7(−3−1)+3(1−4)∣\frac{1}{2}\left|0(4-0)+4(0+3)+7(-3-1)+3(1-4)\right|
Explanation: The skill is finding the perimeter of a quadrilateral using coordinates. The vertices are A(0,1), B(4,4), C(7,0), and D(3,-3). Side lengths are computed using the distance formula between consecutive points. The perimeter is the sum of these four side lengths. Each simplifies to 5, summing to 20, justifying the expression in the correct choice. A distractor misconception is summing only three sides, as in choice C. To transfer this strategy, always compute individual side lengths before summing for the perimeter.

Question 15

Triangle DEFDEF has vertices D(1,2)D(1,2), E(7,2)E(7,2), and F(7,8)F(7,8). Which value correctly represents the area?

  1. 1818 (correct answer)
  2. 3636
  3. 1212
  4. 2424
Explanation: This problem asks us to find the area of a triangle using coordinates. The vertices are D(1,2), E(7,2), and F(7,8). Since D and E have the same y-coordinate (2), DE is horizontal with length |7-1| = 6. Since E and F have the same x-coordinate (7), EF is vertical with length |8-2| = 6. This forms a right triangle with legs of length 6 each, so the area is (1/2) × 6 × 6 = 18. A common error is computing the full rectangle area (36) without dividing by 2. For right triangles with coordinate vertices, use the two perpendicular sides as base and height.

Question 16

Quadrilateral WXYZ has vertices W(1,−2)W(1,-2), X(5,0)X(5,0), Y(3,4)Y(3,4), and Z(−1,2)Z(-1,2) connected in that order. Which expression represents the perimeter?

  1. (5−1)2+(0+2)2+(3−5)2+(4−0)2+(−1−3)2+(2−4)2+(1+1)2+(−2−2)2\sqrt{(5-1)^2+(0+2)^2}+\sqrt{(3-5)^2+(4-0)^2}+\sqrt{(-1-3)^2+(2-4)^2}+\sqrt{(1+1)^2+(-2-2)^2} (correct answer)
  2. (5−1)2+(0+2)2+(3−5)2+(4−0)2+(−1−3)2+(2−4)2\sqrt{(5-1)^2+(0+2)^2}+\sqrt{(3-5)^2+(4-0)^2}+\sqrt{(-1-3)^2+(2-4)^2}
  3. (5−1)+(4−0)+(3−(−1))+(2−(−2))(5-1)+(4-0)+(3-(-1))+(2-(-2))
  4. 12∣1(0−4)+5(4−2)+3(2+2)+(−1)(−2−0)∣\frac{1}{2}\left|1(0-4)+5(4-2)+3(2+2)+(-1)(-2-0)\right|
Explanation: The skill is finding the perimeter of a quadrilateral using coordinates. The vertices are W(1,-2), X(5,0), Y(3,4), and Z(-1,2). Side lengths are computed using the distance formula between consecutive points. The perimeter is the sum of all four side lengths. Each side calculates to √20, but the expression summing the four distance formulas justifies the perimeter. A distractor misconception is omitting one side, as in choice B with only three terms. To transfer this strategy, always compute individual side lengths before summing for the perimeter.

Question 17

Triangle PQR is drawn on a coordinate plane with vertices P(−1,2)P(-1,2), Q(4,1)Q(4,1), and R(2,5)R(2,5). What is the area of the triangle?

  1. 12⋅5⋅3=7.5\frac{1}{2}\cdot 5\cdot 3=7.5
  2. 12∣(−1)(1−5)+4(5−2)+2(2−1)∣=9\frac{1}{2}\left|(-1)(1-5)+4(5-2)+2(2-1)\right|=9 (correct answer)
  3. (4+1)2+(1−2)2+(2−4)2+(5−1)2+(−1−2)2+(2−5)2\sqrt{(4+1)^2+(1-2)^2}+\sqrt{(2-4)^2+(5-1)^2}+\sqrt{(-1-2)^2+(2-5)^2}
  4. 12∣(−1)(1−5)+4(2−5)+2(5−1)∣=3\frac{1}{2}\left|(-1)(1-5)+4(2-5)+2(5-1)\right|=3
Explanation: The skill is finding the area of a triangle using coordinates. The vertices are P(-1,2), Q(4,1), and R(2,5). Side lengths are computed using the distance formula, but for area, the shoelace formula is efficient. The area logic involves listing coordinates and computing half the absolute value of the summed products as shown. This yields 9, justifying the value in the correct choice. A distractor misconception is confusing area with perimeter, as in choice C which sums distances. To transfer this strategy, compute side lengths if needed, but use shoelace directly for area.

Question 18

Triangle LMN has vertices L(−2,0)L(-2,0), M(2,0)M(2,0), and N(1,3)N(1,3) on the coordinate plane. Which value correctly represents the area?​

  1. 12⋅4⋅3=6\frac{1}{2}\cdot 4\cdot 3=6 (correct answer)
  2. 4+(1−2)2+(3−0)2+(1+2)2+(3−0)24+\sqrt{(1-2)^2+(3-0)^2}+\sqrt{(1+2)^2+(3-0)^2}
  3. 12⋅3⋅3=4.5\frac{1}{2}\cdot 3\cdot 3=4.5
  4. 12⋅2⋅3=3\frac{1}{2}\cdot 2\cdot 3=3
Explanation: The skill is finding the area of a triangle using coordinates. The vertices are L(-2,0), M(2,0), and N(1,3). Side lengths are computed using the distance formula, but base and height are evident. The area logic is half base times height. With base 4 and height 3, this gives 6, justifying the value in the correct choice. A distractor misconception is using incorrect height, as in choice D with 2. To transfer this strategy, identify base and height or use shoelace after noting coordinates.

Question 19

Quadrilateral KLMNKLMN has vertices K(−1,1)K(-1,1), L(2,2)L(2,2), M(4,−1)M(4,-1), and N(1,−2)N(1,-2) in that order. Which expression represents the perimeter?

  1. 10+13+10+13\sqrt{10}+\sqrt{13}+\sqrt{10}+\sqrt{13} (correct answer)
  2. 10+13+10\sqrt{10}+\sqrt{13}+\sqrt{10}
  3. 3+3+3+33+3+3+3
  4. 10+13+10+1310+13+10+13
Explanation: The skill is finding the perimeter of a quadrilateral using coordinates. The vertices are K(-1,1), L(2,2), M(4,-1), and N(1,-2). Side lengths are computed using the distance formula, which is the square root of the sum of the squares of the differences in x and y coordinates. The perimeter is the sum of all four side lengths. Calculating each distance gives KL = √10, LM = √13, MN = √10, and NK = √13, justifying the final value as √10 + √13 + √10 + √13. A common distractor misconception is adding without square roots, leading to 10 + 13 + 10 + 13. To transfer this strategy, compute side lengths before summing for perimeter or using them for area in other figures.

Question 20

Triangle ABCABC has vertices A(1,1)A(1,1), B(7,1)B(7,1), and C(4,5)C(4,5) in the coordinate plane (units are uniform). Which value correctly represents the area?

  1. 1212 (correct answer)
  2. 2424
  3. 1010
  4. 1414
Explanation: This problem asks for the area of triangle ABC with vertices A(1,1), B(7,1), and C(4,5). The vertices form a triangle with A(1,1), B(7,1), and C(4,5). Since A and B have the same y-coordinate (y=1), AB is horizontal with length |7-1| = 6. The height from C to line AB is the vertical distance |5-1| = 4. For a triangle with a horizontal base, Area = ½ × base × height = ½ × 6 × 4 = 12. A common error is forgetting the ½ factor, which would give 24 instead of 12. The transfer strategy is to look for horizontal or vertical sides, which simplify area calculations.