12 multiple-choice questions, progressively harder.
Points AAA and BBB lie on the same horizontal line, 888 units apart. If A=(−3,5)A = (-3, 5)A=(−3,5) and BBB is to the right of AAA, what is BBB?
Solution
Correct answer: C
On the same horizontal line, BBB has the same y-coordinate as AAA, namely 555. Moving 888 units right adds 888 to the x-coordinate.
x=−3+8=5 ⇒ B=(5,5).x = -3 + 8 = 5 \;\Rightarrow\; B = (5, 5).x=−3+8=5⇒B=(5,5).
Which point is the reflection of (−3,−6)(-3, -6)(−3,−6) across the x-axis?
Correct answer: A
Reflecting across the x-axis keeps the x-coordinate and flips the sign of the y-coordinate.
(−3,−6)→(−3,6).(-3, -6) \to (-3, 6).(−3,−6)→(−3,6).
A rectangle has corners at (−4,−2)(-4, -2)(−4,−2), (3,−2)(3, -2)(3,−2), (3,2)(3, 2)(3,2), and (−4,2)(-4, 2)(−4,2). What is its area?
Correct answer: B
Find a horizontal side from (−4,−2)(-4, -2)(−4,−2) to (3,−2)(3, -2)(3,−2) and a vertical side from (−4,−2)(-4, -2)(−4,−2) to (−4,2)(-4, 2)(−4,2).
l=∣3−(−4)∣=7,w=∣2−(−2)∣=4.l = |3 - (-4)| = 7, \qquad w = |2 - (-2)| = 4.l=∣3−(−4)∣=7,w=∣2−(−2)∣=4.
Then the area is
A=7×4=28.A = 7 \times 4 = 28.A=7×4=28.
The point (a,b)(a, b)(a,b) lies in Quadrant II. Which statement is always true?
Correct answer: D
Quadrant II is the upper-left region: left of the y-axis means a negative x-coordinate, above the x-axis means a positive y-coordinate.
Quadrant II ⇒ a<0, b>0.\text{Quadrant II} \;\Rightarrow\; a < 0, \; b > 0.Quadrant II⇒a<0,b>0.
Points (x,4)(x, 4)(x,4) and (9,4)(9, 4)(9,4) are 666 units apart, and the first is to the left of the second. What is xxx?
The points share y=4y = 4y=4, so the distance is the gap in the x-coordinates. The first is 666 units left of x=9x = 9x=9.
9−6=3 ⇒ x=3.9 - 6 = 3 \;\Rightarrow\; x = 3.9−6=3⇒x=3.
Check: ∣9−3∣=6|9 - 3| = 6∣9−3∣=6.
A point in Quadrant I is reflected across the x-axis. In which quadrant is its image?
A Quadrant I point has the pattern (+,+)(+, +)(+,+). Reflecting across the x-axis flips the sign of the y-coordinate, giving (+,−)(+, -)(+,−).
(+,+)→(+,−) ⇒ Quadrant IV.(+, +) \to (+, -) \;\Rightarrow\; \text{Quadrant IV}.(+,+)→(+,−)⇒Quadrant IV.
A rectangle has corners at (−5,3)(-5, 3)(−5,3), (−5,−2)(-5, -2)(−5,−2), (1,−2)(1, -2)(1,−2), and (1,3)(1, 3)(1,3). What is its perimeter?
Find a vertical side from (−5,3)(-5, 3)(−5,3) to (−5,−2)(-5, -2)(−5,−2) and a horizontal side from (−5,−2)(-5, -2)(−5,−2) to (1,−2)(1, -2)(1,−2).
w=∣3−(−2)∣=5,l=∣1−(−5)∣=6.w = |3 - (-2)| = 5, \qquad l = |1 - (-5)| = 6.w=∣3−(−2)∣=5,l=∣1−(−5)∣=6.
Then the perimeter is
P=2(6+5)=22.P = 2(6 + 5) = 22.P=2(6+5)=22.
Three corners of a rectangle are (1,1)(1, 1)(1,1), (7,1)(7, 1)(7,1), and (7,4)(7, 4)(7,4). What is the fourth corner?
In an axis-aligned rectangle the fourth corner shares its x-coordinate with (1,1)(1, 1)(1,1) and its y-coordinate with (7,4)(7, 4)(7,4).
( x=1, y=4 ) ⇒ (1,4).(\,x = 1, \; y = 4\,) \;\Rightarrow\; (1, 4).(x=1,y=4)⇒(1,4).
That completes the rectangle.
Point P=(−4,6)P = (-4, 6)P=(−4,6) is reflected across the x-axis to give P′P'P′, and then P′P'P′ is reflected across the y-axis to give P′′P''P′′. What is P′′P''P′′?
Reflecting (−4,6)(-4, 6)(−4,6) across the x-axis flips the y-sign: P′=(−4,−6)P' = (-4, -6)P′=(−4,−6). Reflecting P′P'P′ across the y-axis flips the x-sign:
P′′=(4,−6).P'' = (4, -6).P′′=(4,−6).
Each reflection flipped exactly one coordinate's sign.
The point (0,k)(0, k)(0,k) lies on an axis for every value of kkk. Which axis (and what is the special case)?
An x-coordinate of 000 puts the point on the y-axis for any kkk.
(0,k) ⇒ on the y-axis.(0, k) \;\Rightarrow\; \text{on the y-axis}.(0,k)⇒on the y-axis.
The one special case is k=0k = 0k=0, which is the origin, the point shared by both axes.
Two opposite corners of an axis-aligned rectangle are (−2,−1)(-2, -1)(−2,−1) and (4,3)(4, 3)(4,3). What is its area?
The horizontal side spans the x-coordinates and the vertical side spans the y-coordinates of the two opposite corners.
l=∣4−(−2)∣=6,w=∣3−(−1)∣=4.l = |4 - (-2)| = 6, \qquad w = |3 - (-1)| = 4.l=∣4−(−2)∣=6,w=∣3−(−1)∣=4.
So the area is
A=6×4=24.A = 6 \times 4 = 24.A=6×4=24.
Point EEE lies in Quadrant IV, 555 units from the y-axis and 222 units from the x-axis. What are its coordinates?
Distance from the y-axis is the size of the x-coordinate, and distance from the x-axis is the size of the y-coordinate. Quadrant IV has the pattern (+,−)(+, -)(+,−).
∣x∣=5, ∣y∣=2, (+,−) ⇒ (5,−2).|x| = 5, \; |y| = 2, \; (+, -) \;\Rightarrow\; (5, -2).∣x∣=5,∣y∣=2,(+,−)⇒(5,−2).
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