12 multiple-choice questions, progressively harder.
What are the coordinates of point TTT shown on the grid?
Solution
Correct answer: A
Drop from TTT to the x-axis: it sits at 444. Slide across to the y-axis: it sits at −1-1−1.
T=(4,−1).T = (4, -1).T=(4,−1).
A rectangle has corners at (−1,−1)(-1, -1)(−1,−1), (5,−1)(5, -1)(5,−1), (5,6)(5, 6)(5,6), and (−1,6)(-1, 6)(−1,6). What is its area?
Correct answer: C
Find a horizontal side from (−1,−1)(-1, -1)(−1,−1) to (5,−1)(5, -1)(5,−1) and a vertical side from (−1,−1)(-1, -1)(−1,−1) to (−1,6)(-1, 6)(−1,6).
l=∣5−(−1)∣=6,w=∣6−(−1)∣=7.l = |5 - (-1)| = 6, \qquad w = |6 - (-1)| = 7.l=∣5−(−1)∣=6,w=∣6−(−1)∣=7.
Then the area is
A=6×7=42.A = 6 \times 7 = 42.A=6×7=42.
A point (x,y)(x, y)(x,y) has x<0x < 0x<0 and y<0y < 0y<0. Where is it?
Correct answer: D
Both coordinates are negative, the lower-left region.
(−,−) ⇒ Quadrant III.(-, -) \;\Rightarrow\; \text{Quadrant III}.(−,−)⇒Quadrant III.
Points (−7,2)(-7, 2)(−7,2) and (−7,−6)(-7, -6)(−7,−6) share a coordinate. How far apart are they?
Correct answer: B
Both points have x=−7x = -7x=−7, so the segment is vertical. Subtract the y-coordinates and take the absolute value.
∣2−(−6)∣=∣2+6∣=8.|2 - (-6)| = |2 + 6| = 8.∣2−(−6)∣=∣2+6∣=8.
They are 888 units apart.
Three corners of an axis-aligned rectangle are (−2,1)(-2, 1)(−2,1), (−2,5)(-2, 5)(−2,5), and (4,5)(4, 5)(4,5). What is the fourth corner?
The fourth corner shares its x-coordinate with (4,5)(4, 5)(4,5) and its y-coordinate with (−2,1)(-2, 1)(−2,1).
( x=4, y=1 ) ⇒ (4,1).(\,x = 4, \; y = 1\,) \;\Rightarrow\; (4, 1).(x=4,y=1)⇒(4,1).
Which point lies 333 units above (2,−5)(2, -5)(2,−5)?
Moving up changes only the y-coordinate, adding 333.
y=−5+3=−2 ⇒ (2,−2).y = -5 + 3 = -2 \;\Rightarrow\; (2, -2).y=−5+3=−2⇒(2,−2).
The point (−9,0)(-9, 0)(−9,0) lies where?
Its y-coordinate is 000, so it lies on the horizontal axis.
y=0 ⇒ on the x-axis.y = 0 \;\Rightarrow\; \text{on the x-axis}.y=0⇒on the x-axis.
A point on an axis is in no quadrant.
A square has a vertical side from (3,−1)(3, -1)(3,−1) to (3,5)(3, 5)(3,5). What is its perimeter?
The given side has length
∣5−(−1)∣=6.|5 - (-1)| = 6.∣5−(−1)∣=6.
A square has four equal sides, so its perimeter is
P=4×6=24.P = 4 \times 6 = 24.P=4×6=24.
Two opposite corners of an axis-aligned rectangle are (−3,−2)(-3, -2)(−3,−2) and (2,4)(2, 4)(2,4). What is its perimeter?
The side lengths come from the gaps in the x-coordinates and the y-coordinates.
l=∣2−(−3)∣=5,w=∣4−(−2)∣=6.l = |2 - (-3)| = 5, \qquad w = |4 - (-2)| = 6.l=∣2−(−3)∣=5,w=∣4−(−2)∣=6.
Then the perimeter is
P=2(5+6)=22.P = 2(5 + 6) = 22.P=2(5+6)=22.
Point BBB lies in Quadrant I, 666 units from the y-axis and 333 units from the x-axis. What are its coordinates?
Distance from the y-axis is ∣x∣|x|∣x∣ and distance from the x-axis is ∣y∣|y|∣y∣. Quadrant I has both coordinates positive.
∣x∣=6, ∣y∣=3, (+,+) ⇒ (6,3).|x| = 6, \; |y| = 3, \; (+, +) \;\Rightarrow\; (6, 3).∣x∣=6,∣y∣=3,(+,+)⇒(6,3).
A rectangle has a horizontal side from (−6,2)(-6, 2)(−6,2) to (3,2)(3, 2)(3,2) and a vertical side of length 444. What is its perimeter?
The horizontal side has length
∣3−(−6)∣=9,|3 - (-6)| = 9,∣3−(−6)∣=9,
and the vertical side is 444. Add a length and a width, then double:
P=2(9+4)=2×13=26.P = 2(9 + 4) = 2 \times 13 = 26.P=2(9+4)=2×13=26.
How far is the point (−8,5)(-8, 5)(−8,5) from the y-axis?
The distance from the y-axis is the horizontal distance, which is the size of the x-coordinate.
∣−8∣=8.|-8| = 8.∣−8∣=8.
The point is 888 units from the y-axis.
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