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Geometry Help: Coordinates To Find Perimeter And Area

Review real example questions for Coordinates To Find Perimeter And Area in Geometry.

Question 1 / 10

0 of 10 answered

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?

All questions

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?

27 square units (correct answer)
30 square units
33 square units
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'?

0 (correct answer)
4
8
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 $ABCD$ has vertices $A(0,0)$ , $B(4,0)$ , $C(3,3)$ , and $D(-1,3)$ in that order. What is the perimeter of the figure?

$14$
$4+3+4+3$
$4+\sqrt{10}+4+\sqrt{10}$ (correct answer)
$4+\sqrt{18}+4+\sqrt{18}$

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?

24 square units
36 square units
48 square units (correct answer)
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 $A(-1,2)$ , $B(3,1)$ , and $C(1,5)$ are plotted on a coordinate plane and connected to form triangle $ABC$ (segments $\overline{AB}$ , $\overline{BC}$ , and $\overline{CA}$ ). What is the perimeter of the figure?

$4+\sqrt{17}+\sqrt{13}$
$\sqrt{17}+\sqrt{20}+\sqrt{13}$ (correct answer)
$\sqrt{17}+\sqrt{20}$
$17+20+13$

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'?

9:4
3:2
5:1
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 $1:1$ . Looking at the incorrect choices: Answer A) $9:4$ might tempt you if you incorrectly squared the translation distances (3² and 2²), but translation distances don't affect area ratios. Answer B) $3:2$ 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) $5:1$ 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 $1:1$ .

Question 7

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

$25$
$20$ (correct answer)
$50$
$10$

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 $ABCDE$ has vertices $A(0,0)$ , $B(6,0)$ , $C(6,4)$ , $D(3,7)$ , and $E(0,4)$ . What is the area of the pentagon?

$27$ square units
$33$ square units (correct answer)
$36$ square units
$42$ square units

Explanation: Decompose into rectangle $ABCE$ with area $6 \times 4 = 24$ and triangle $CDE$ with base $CE = 6$ and height $= 7-4 = 3$ , area $= \frac{1}{2}(6)(3) = 9$ . Total $= 33$ . 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 $JKLM$ has vertices $J(-4, -1)$ , $K(-1, 3)$ , $L(4, 3)$ , and $M(1, -1)$ . What is the perimeter of $JKLM$ ?

$16$ units
$18$ units
$20$ units (correct answer)
$25$ units

Explanation: $JK = \sqrt{9+16} = 5$ ; $KL = 5$ (horizontal); $LM = \sqrt{9+16} = 5$ ; $MJ = 5$ (horizontal). Perimeter $= 20$ . 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?

12 square units
15 square units (correct answer)
18 square units
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.