12 multiple-choice questions, progressively harder.
What are the coordinates of point KKK shown on the grid?
Solution
Correct answer: D
Drop from KKK to the x-axis: it sits at −2-2−2. Slide across to the y-axis: it sits at 555.
K=(−2,5).K = (-2, 5).K=(−2,5).
Reflect (5,−8)(5, -8)(5,−8) across the y-axis. What is the image?
Correct answer: C
Reflecting across the y-axis keeps the y-coordinate and flips the sign of the x-coordinate.
(5,−8)→(−5,−8).(5, -8) \to (-5, -8).(5,−8)→(−5,−8).
A rectangle has corners at (2,1)(2, 1)(2,1), (8,1)(8, 1)(8,1), (8,5)(8, 5)(8,5), and (2,5)(2, 5)(2,5). What is its perimeter?
Correct answer: B
Find a horizontal side from (2,1)(2, 1)(2,1) to (8,1)(8, 1)(8,1) and a vertical side from (2,1)(2, 1)(2,1) to (2,5)(2, 5)(2,5).
l=∣8−2∣=6,w=∣5−1∣=4.l = |8 - 2| = 6, \qquad w = |5 - 1| = 4.l=∣8−2∣=6,w=∣5−1∣=4.
Then apply the perimeter formula:
P=2(6+4)=20.P = 2(6 + 4) = 20.P=2(6+4)=20.
Point A=(−3,4)A = (-3, 4)A=(−3,4) is reflected across the x-axis. In which quadrant is the image?
AAA is in Quadrant II, pattern (−,+)(-, +)(−,+). Reflecting across the x-axis flips the y-sign to give (−,−)(-, -)(−,−).
(−,+)→(−,−) ⇒ Quadrant III.(-, +) \to (-, -) \;\Rightarrow\; \text{Quadrant III}.(−,+)→(−,−)⇒Quadrant III.
Which point shown on the grid lies in Quadrant III?
Quadrant III is the lower-left region, where both coordinates are negative. Point YYY sits there, at (−2,−3)(-2, -3)(−2,−3).
Y=(−2,−3): (−,−) ⇒ Quadrant III.Y = (-2, -3): \; (-, -) \;\Rightarrow\; \text{Quadrant III}.Y=(−2,−3):(−,−)⇒Quadrant III.
A rectangle has corners at (0,0)(0, 0)(0,0), (7,0)(7, 0)(7,0), (7,5)(7, 5)(7,5), and (0,5)(0, 5)(0,5). What is its area?
Correct answer: A
The bottom side from (0,0)(0, 0)(0,0) to (7,0)(7, 0)(7,0) has length 777, and the left side from (0,0)(0, 0)(0,0) to (0,5)(0, 5)(0,5) has length 555.
A=7×5=35.A = 7 \times 5 = 35.A=7×5=35.
A square has one horizontal side from (1,2)(1, 2)(1,2) to (6,2)(6, 2)(6,2). What is the area of the square?
The given side runs from (1,2)(1, 2)(1,2) to (6,2)(6, 2)(6,2), so its length is
∣6−1∣=5.|6 - 1| = 5.∣6−1∣=5.
A square has all sides equal, so each side is 555 and the area is
A=5×5=25.A = 5 \times 5 = 25.A=5×5=25.
Starting at (1,−2)(1, -2)(1,−2), you move 333 left and 555 up. In which quadrant do you land?
Moving 333 left gives x=1−3=−2x = 1 - 3 = -2x=1−3=−2, and moving 555 up gives y=−2+5=3y = -2 + 5 = 3y=−2+5=3, so the point is (−2,3)(-2, 3)(−2,3).
(−,+) ⇒ Quadrant II.(-, +) \;\Rightarrow\; \text{Quadrant II}.(−,+)⇒Quadrant II.
If both coordinates of a point have the same sign (both positive or both negative), which two quadrants could it be in?
Both positive, (+,+)(+, +)(+,+), is Quadrant I; both negative, (−,−)(-, -)(−,−), is Quadrant III.
(+,+)→I,(−,−)→III.(+, +) \to \text{I}, \qquad (-, -) \to \text{III}.(+,+)→I,(−,−)→III.
So same-sign coordinates fall in Quadrant I or Quadrant III.
Point (m,n)(m, n)(m,n) lies in Quadrant IV. What is true of the reflection of (m,n)(m, n)(m,n) across the y-axis?
A Quadrant IV point has the pattern (+,−)(+, -)(+,−). Reflecting across the y-axis flips the x-sign to give (−,−)(-, -)(−,−).
(+,−)→(−,−) ⇒ Quadrant III.(+, -) \to (-, -) \;\Rightarrow\; \text{Quadrant III}.(+,−)→(−,−)⇒Quadrant III.
A rectangle on the grid has a horizontal side from (−3,−1)(-3, -1)(−3,−1) to (5,−1)(5, -1)(5,−1) and a vertical side of length 333. What is its area?
The horizontal side has length
∣5−(−3)∣=8,|5 - (-3)| = 8,∣5−(−3)∣=8,
and the vertical side is given as 333, so the area is
A=8×3=24.A = 8 \times 3 = 24.A=8×3=24.
A rectangle has corners at (−5,−3)(-5, -3)(−5,−3), (2,−3)(2, -3)(2,−3), (2,4)(2, 4)(2,4), and (−5,4)(-5, 4)(−5,4). Is it a square, and what is its area?
Measure one horizontal and one vertical side.
l=∣2−(−5)∣=7,w=∣4−(−3)∣=7.l = |2 - (-5)| = 7, \qquad w = |4 - (-3)| = 7.l=∣2−(−5)∣=7,w=∣4−(−3)∣=7.
The two side lengths are equal, so it is a square, and its area is
A=7×7=49.A = 7 \times 7 = 49.A=7×7=49.
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