Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Read the ordered pair in order: the first number tells you left/right, the second tells you up/down.
Step-by-step solution
Start at the origin (0, 0), where the axes cross.
The x-coordinate is -4. Negative means move left, so move 4 units left along the x-axis.
The y-coordinate is 2. Positive means move up, so from there move 2 units up.
You land in the upper-left region, Quadrant II.
The point is 4 left, 2 up from the origin: (-4,\ 2).
Check: In Quadrant II every point has a negative x and a positive y — (-4, 2) fits: (-,\ +) ✓
Tip: Think "walk before you climb": always move along the x-axis first, then up or down.
Question 2 of 36
No calculator
In which quadrant is the point (-3,\ -5) located?
Hint
Look at the sign of each coordinate, then picture where left/right and up/down send you.
Step-by-step solution
The x-coordinate is -3: negative, so the point is to the left of the y-axis.
The y-coordinate is -5: negative, so the point is below the x-axis.
Left and below is the lower-left region.
The quadrants are numbered counter-clockwise starting at the upper right: I (upper right), II (upper left), III (lower left), IV (lower right).
So (-3, -5) is in Quadrant III.
Check: Quadrant III is the only quadrant with sign pattern (-,\ -) ✓
Tip: Memorize the sign patterns: I (+,+), II (-,+), III (-,-), IV (+,-).
Question 3 of 36
Use the coordinate plane below.
What are the coordinates of point B?
Hint
Drop straight down (or up) from the point to the x-axis to read x. Then go across to the y-axis to read y.
Step-by-step solution
Find point B in the upper-left part of the grid.
Go straight down from B to the x-axis: you reach the tick marked -4. So x = -4.
Go straight across from B to the y-axis: you reach the tick marked 2. So y = 2.
Write the pair as (x,\ y).
Point B is (-4,\ 2).
Check: B is left of the y-axis (negative x) and above the x-axis (positive y) — Quadrant II, sign pattern (-, +) ✓
Tip: The order is always (x,\ y) — "across, then up/down", like the alphabet: x comes before y.
Question 4 of 36
No calculator
A point lies on the y-axis, 3 units below the origin. Which ordered pair names this point?
Hint
Every point on the y-axis has the same x-coordinate. What is it?
Step-by-step solution
A point on the y-axis has not moved left or right at all, so its x-coordinate is 0.
"3 units below the origin" means the y-coordinate is -3 (down is negative).
Put them in order (x,\ y).
The point is (0,\ -3).
Check: Plot it: start at the origin, move 0 left/right, then 3 down — you are on the vertical axis ✓
Tip: On the y-axis, x = 0. On the x-axis, y = 0. Points on an axis are not in any quadrant.
Question 5 of 36
No calculator
A delivery app places its warehouse at the origin of a coordinate grid. A grocery store is at the point (7,\ -4).
In which quadrant is the grocery store?
Hint
Look at the sign of each coordinate: is x positive or negative? Is y positive or negative?
Step-by-step solution
Read the ordered pair (x, y) = (7, -4). The first number is x and the second is y.
x = 7 is positive, so the point is to the right of the y-axis.
y = -4 is negative, so the point is below the x-axis.
Right and below is the lower-right region. The quadrants are numbered counterclockwise starting at the upper right: I (upper right), II (upper left), III (lower left), IV (lower right).
The store is in Quadrant IV.
Check: The sign pattern of Quadrant IV is (+,\ -), and (7, -4) has exactly that pattern ✓
Tip: Memorize the sign patterns: I (+,+), II (-,+), III (-,-), IV (+,-). Quadrants go counterclockwise, like the letter C drawn backwards from the top right.
Question 6 of 36
Hot spot
A robot vacuum's charging dock is marked on a floor plan drawn on a coordinate grid. The dock is at (3,\ -4).
Click the location of the dock.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Start at the origin. The first number tells you how far to move left or right; the second tells you up or down.
Step-by-step solution
Start at the origin (0, 0).
The x-coordinate is 3 (positive), so move 3 units right along the x-axis.
The y-coordinate is -4 (negative), so from there move 4 units down.
Mark the point where you stop: 3 right, 4 down.
The dock is at (3, -4), in the lower-right part of the grid (Quadrant IV).
Check: Read the point back: from the origin it is 3 units to the right (x = 3) and 4 units below the x-axis (y = −4) ✓
Tip: Always move across first, then up/down — just like reading: left-to-right comes before top-to-bottom. "Run before you jump."
Question 7 of 36
No calculator
Which of these points lies on the x-axis?
Hint
Every point on the x-axis is at the same height as the origin. What must its y-coordinate be?
Step-by-step solution
The x-axis is the horizontal number line through the origin.
A point on it is neither above nor below the origin, so its height — the y-coordinate — must be 0.
So we want a point of the form (\text{number},\ 0).
Check each choice:
(0, 6): y = 6, so it is 6 up — this is on the y-axis.
(6, 0): y = 0 ✓ — on the x-axis, 6 units right of the origin.
(6, 6): y = 6, inside Quadrant I.
(-6, 6): y = 6, inside Quadrant II.
The answer is (6, 0).
Check: Plot (6, 0): move 6 right and 0 up — you are still sitting on the horizontal axis ✓
Tip: On the x-axis, y = 0. On the y-axis, x = 0. The coordinate that is zero is the other letter.
Question 8 of 36
No calculator
Six points are labeled with letters on the grid.
Which ordered pair gives the location of point R?
Hint
From the origin, count how far left or right you go to get under R, then how far up or down.
Step-by-step solution
Find point R in the lower-left part of the grid.
Go straight up from R to the x-axis: you reach the tick mark -4. So x = -4 (4 units left).
Go straight across from R to the y-axis: you reach -3. So y = -3 (3 units down).
Write x first: (x, y) = (-4, -3).
Point R is at (-4, -3).
Check: R is in the lower-left region, Quadrant III, where both coordinates are negative — and both of ours are negative ✓
Tip: Read a point like an address: first the "street" (left/right, x), then the "floor" (up/down, y). Always check that the signs match the quadrant.
Question 9 of 36
No calculator
In a board game drawn on a coordinate grid, a game piece starts at (-1,\ 4). It moves 6 units down and then 2 units left.
Where does the piece end up?
Hint
Left and right change only the x-coordinate. Up and down change only the y-coordinate.
Step-by-step solution
Start: (x, y) = (-1, 4).
"6 units down" changes only y, and down means subtract:
y = 4 - 6 = -2
"2 units left" changes only x, and left means subtract:
x = -1 - 2 = -3
Put them together: (-3, -2).
The piece ends at (-3, -2).
Check: Go backwards from (-3, -2): 2 right gives x = -1, 6 up gives y = 4 — back to (-1, 4) ✓
Tip: Right = + to x, left = - from x, up = + to y, down = - from y. The order of the moves does not change where you finish.
Question 10 of 36
No calculatorSelect all
Select all the points that lie in Quadrant II.
Select all that apply.
Hint
Quadrant II is the upper-left region. Which sign must x have there? Which sign must y have?
Step-by-step solution
Quadrant II is up and to the left, so a point there needs x < 0 and y > 0. Test each one:
(-2,\ 5): x = -2 < 0, y = 5 > 0 → Quadrant II ✓
(5,\ -2): x > 0, y < 0 → Quadrant IV ✗
(-3,\ -3): both negative → Quadrant III ✗
(0,\ 4): x = 0 → on the y-axis, not in any quadrant ✗
(-7,\ 1): x = -7 < 0, y = 1 > 0 → Quadrant II ✓
The answers are (-2,\ 5) and (-7,\ 1).
Check: Both chosen points have the pattern (-,\ +); none of the others do ✓
Tip: Don't be fooled by size: (-7, 1) is far left and barely up, but it is still in Quadrant II. Only the signs matter.
Question 11 of 36
Fill in
A town map is drawn on a coordinate grid where each unit is one city block. The library is at (-3,\ 2) and the park is at (4,\ 2). Streets run only along grid lines.
How many blocks is the shortest walk from the library to the park?
blocks
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The two places have the same y-coordinate, so the walk is a straight horizontal line. How far apart are the x-coordinates?
Step-by-step solution
Both points have y = 2, so they are on the same horizontal line. Walk straight east.
Horizontal distance = difference of the x-coordinates:
4 - (-3) = 4 + 3 = 7
Another way to see it: from x = -3 to x = 0 is 3 blocks, and from x = 0 to x = 4 is 4 more blocks: 3 + 4 = 7.
The walk is 7 blocks.
Check: Count on the grid: -3 \to -2 \to -1 \to 0 \to 1 \to 2 \to 3 \to 4 is 7 steps ✓
Tip: When a trip crosses an axis, distances on the two sides add. Subtracting 4 - 3 = 1 forgets that -3 is on the other side of zero.
Question 12 of 36
No calculator
The point (a,\ b) lies in Quadrant IV. Which statement must be true?
Hint
Picture Quadrant IV: is it left or right of the y-axis? Above or below the x-axis?
Step-by-step solution
Quadrant IV is the lower-right region of the plane.
"Right of the y-axis" means the x-coordinate is positive: a > 0.
"Below the x-axis" means the y-coordinate is negative: b < 0.
So a > 0 and b < 0.
Check: Try an example in Quadrant IV, like (3, -2): a = 3 > 0 and b = -2 < 0 ✓. None of the other choices works for (3, -2).
Tip: Turn quadrant questions into sign questions. Writing I (+,+), II (-,+), III (-,-), IV (+,-) in the corner of your erasable note board takes five seconds.
Question 13 of 36
No calculator
The point (5,\ -1) is reflected across the x-axis. What are the coordinates of the new point?
Hint
A reflection across the x-axis is a flip over the horizontal axis. Which coordinate changes, and which stays the same?
Step-by-step solution
Reflecting across the x-axis flips the point up ↔ down. Its left/right position does not change.
So the x-coordinate stays 5.
The y-coordinate changes sign: -1 becomes 1.
The point was 1 unit below the axis; its mirror image is 1 unit above.
The new point is (5,\ 1).
Check: The original (5, -1) and the image (5, 1) are the same distance (1 unit) from the x-axis, on opposite sides ✓
Tip: Across the x-axis: (x, y) \to (x, -y). Across the y-axis: (x, y) \to (-x, y). The axis you reflect across names the coordinate that stays.
Question 14 of 36
Hot spot
Point P is at (-2,\ 1). Click the point that is 5 units to the right of P and 3 units down from P.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Moving right or left changes only x. Moving up or down changes only y.
Step-by-step solution
Start at P(-2,\ 1).
"5 units right" adds 5 to x: -2 + 5 = 3.
"3 units down" subtracts 3 from y: 1 - 3 = -2.
The new point is (3,\ -2), in Quadrant IV.
Click (3,\ -2).
Check: From (3, -2) go back 5 left and 3 up: (3 - 5,\ -2 + 3) = (-2,\ 1) = P ✓
Tip: Right = +x, left = -x, up = +y, down = -y. Do the arithmetic on your erasable note board before you click — especially when you cross zero.
Question 15 of 36
No calculator
For the point (a,\ b), you know that a < 0 and b > 0.
In which quadrant is the point (a,\ b)?
Hint
Translate each inequality into a direction: is the point left or right of the y-axis? Above or below the x-axis?
Step-by-step solution
a is the x-coordinate. a < 0 means a is negative, so the point is left of the y-axis.
b is the y-coordinate. b > 0 means b is positive, so the point is above the x-axis.
Left and above is the upper-left region.
Quadrant numbering (counterclockwise from upper right): I upper right, II upper left, III lower left, IV lower right.
The point is in Quadrant II.
Check: Try real numbers that fit: a = -2, b = 3. The point (-2, 3) is 2 left and 3 up — upper left, Quadrant II ✓
Tip: When a problem uses letters, pick simple numbers that satisfy the conditions (like -2 and 3) and plot them. It turns an abstract question into a picture.
Question 16 of 36
Fill in
A backyard plan is drawn on a coordinate grid where each unit is 1 foot. A sprinkler is at (2,\ -6) and the outdoor faucet is at (2,\ 9).
How many feet long is a straight hose that runs from the sprinkler to the faucet?
feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Both points have the same x-coordinate, so the hose runs straight up and down. How far is it from y = -6 to y = 9?
Step-by-step solution
Compare the points: (2, -6) and (2, 9) have the same x-coordinate, so they lie on one vertical line and the distance is just the change in y.
Subtract the y-coordinates (larger minus smaller):
9 - (-6) = 9 + 6 = 15
Another way to see it: from y = -6 up to 0 is 6 feet, and from 0 up to 9 is 9 feet. 6 + 9 = 15.
The hose is 15 feet long.
Check: Start at y = -6 and add 15: -6 + 15 = 9 — you arrive at the faucet ✓
Tip: When two points share an x (or y) coordinate, the distance is the difference of the other coordinates. Points on opposite sides of an axis: add the distances to the axis — don't subtract 9 - 6 = 3.
Question 17 of 36
No calculator
A quilt pattern is drawn on a coordinate grid. One star is sewn at (-3,\ 4), and the design is mirrored by reflecting it across the y-axis.
Where is the matching star sewn?
Hint
Picture folding the grid along the vertical axis. Which coordinate flips sign, and which stays the same?
Step-by-step solution
The y-axis is the vertical line x = 0. Reflecting across it is like folding the grid along that line.
The point is 3 units left of the y-axis, so its image is 3 units right of it: the x-coordinate changes from -3 to 3.
Folding left-to-right does not move a point up or down, so the y-coordinate stays 4.
Rule: across the y-axis, (x, y) \to (-x, y).
The image is (3, 4).
Check: The midpoint of (-3, 4) and (3, 4) is \left(\tfrac{-3+3}{2}, 4\right) = (0, 4), which lies on the y-axis — the mirror line ✓
Tip: The coordinate that changes is the one that is NOT the axis's name: across the y-axis, x flips; across the x-axis, y flips.
Question 18 of 36
Hot spot
A point has a y-coordinate of 4, and its x-coordinate is the opposite of 2.
Click the point on the coordinate plane.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
First write the point as an ordered pair. What number is the opposite of 2?
Step-by-step solution
The opposite of a number has the same distance from 0 but the other sign. The opposite of 2 is -2.
So x = -2 and y = 4. The point is (-2, 4).
Plot it: from the origin, move 2 units left (because x is negative), then 4 units up (because y is positive).
The point is (-2, 4), in the upper-left region (Quadrant II).
Check: Read the clicked point back: it is 2 units left of the y-axis and 4 units above the x-axis, so x = -2, y = 4, and -2 is indeed the opposite of 2 ✓
Tip: Turn words into an ordered pair before you touch the graph. Write "( \_\_ ,\ \_\_ )" and fill in x first, then y.
Question 19 of 36
No calculator
For a class activity, Maria marked six lettered points on a grid. Her partner calls out the ordered pair (4,\ -3).
Which of Maria's points matches the ordered pair her partner called out?
Hint
Move 4 to the right first, then 3 down. Which letter is there?
Step-by-step solution
The ordered pair (4, -3) means x = 4 and y = -3.
Start at the origin and move 4 units right (positive x).
Then move 3 units down (negative y).
You land in the lower-right region on the point labeled S.
Compare with the other points: P is (-3, 4), Q is (4, 3), R is (-4, -3) — each uses the same digits but a different order or different signs.
The answer is point S.
Check: S is in Quadrant IV, whose sign pattern is (+, -) — matching (4, -3) ✓
Tip: When answer choices use the same digits in different ways, use the signs to pick the quadrant first; that eliminates most choices instantly.
Question 20 of 36
No calculatorFill in
The point (5,\ 3t - 12) lies on the x-axis.
What is the value of t?
t =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
What is true about the y-coordinate of every point on the x-axis?
Step-by-step solution
Every point on the x-axis has a y-coordinate of 0.
The y-coordinate of this point is 3t - 12, so set it equal to 0:
3t - 12 = 0
Add 12 to both sides: 3t = 12.
Divide both sides by 3: t = 4.
The value is t = 4.
Check: With t = 4: 3(4) - 12 = 12 - 12 = 0, so the point is (5, 0), which is on the x-axis ✓
Tip: "On the x-axis" \Rightarrow set y = 0. "On the y-axis" \Rightarrow set x = 0. The x-coordinate (5 here) does not matter at all.
Question 21 of 36
No calculatorSelect all
A scavenger-hunt map marks five clues by their coordinates. Clues hidden in the lower-left section, Quadrant III, are worth double points.
Select all the clue locations that are in Quadrant III.
Select all that apply.
Hint
Quadrant III is the lower-left region. What must be true about the signs of both coordinates?
Step-by-step solution
Quadrant III is the lower-left region: left of the y-axis (x < 0) and below the x-axis (y < 0). Sign pattern (-, -).
Test each point:
(-6, -1): (-, -) ✓ Quadrant III.
(6, -1): (+, -) → Quadrant IV.
(-1.5, -4.5): (-, -) ✓ Quadrant III. Decimals work the same way as whole numbers.
(0, -3): x = 0, so the point sits on the y-axis and is in no quadrant.
(-4, 2): (-, +) → Quadrant II.
The points in Quadrant III are (-6, -1) and (-1.5, -4.5).
Check: Both chosen points are left of and below the origin; none of the others has two negative coordinates ✓
Tip: A coordinate of 0 puts a point on an axis, and axes belong to no quadrant. Watch for this trap in every "select all" quadrant question.
Question 22 of 36
Hot spot
Three corners of a square are at (-4,\ 1), (-1,\ 1) and (-1,\ -2).
Click the location of the fourth corner.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
Plot the three corners first. Which side is missing, and which x and y values does the missing corner share with its neighbors?
Step-by-step solution
Plot (-4, 1), (-1, 1) and (-1, -2).
(-4, 1) and (-1, 1) share y = 1: a horizontal side of length -1 - (-4) = 3.
(-1, 1) and (-1, -2) share x = -1: a vertical side of length 1 - (-2) = 3. Equal sides — good, a square.
The missing corner must be directly below(-4, 1) (same x = -4) and directly left of (-1, -2) (same y = -2).
So the fourth corner is (-4, -2).
The fourth corner is (-4, -2).
Check: Side from (-4, -2) to (-1, -2): length 3; side from (-4, -2) up to (-4, 1): length 3. All four sides are 3 and the sides are horizontal/vertical, so the figure is a square ✓
Tip: For a rectangle or square with sides on grid lines, the missing corner borrows its x from one neighbor and its y from the other.
Question 23 of 36
No calculator
A point was moved 4 units right and 5 units up. After the move, it is at (1,\ -2).
Where was the point before it moved?
Hint
Undo each move: what is the opposite of moving right? Of moving up?
Step-by-step solution
To find the starting point, undo each move by doing the opposite.
The point moved 4 units right, so undo by moving 4 units left (subtract 4 from x):
x = 1 - 4 = -3
The point moved 5 units up, so undo by moving 5 units down (subtract 5 from y):
y = -2 - 5 = -7
The starting point is (-3, -7).
The point started at (-3, -7).
Check: Move (-3, -7) forward: 4 right gives x = 1; 5 up gives y = -2. We land on (1, -2) ✓
Tip: "Work backwards" means reverse each step with the opposite operation. Always check by doing the original moves forward from your answer.
Question 24 of 36
No calculatorDrop-down
The map shows a city park drawn on a coordinate grid.
The Playground is at the point ( , ).
Choose an answer from each menu.
Hint
Find the Playground, then read straight up to the x-axis and straight across to the y-axis.
Step-by-step solution
Find the Playground in the lower-right part of the map.
Read straight up from it to the x-axis: it lines up with 2. So x = 2.
Read straight across to the y-axis: it lines up with -4. So y = -4.
Write x first: (2, -4).
The Playground is at (2, -4).
Check: From the origin, 2 right and 4 down lands on the Playground. Its quadrant is IV, sign pattern (+, -) ✓
Tip: On map questions, labels can sit a little away from the dot. Always read the coordinates of the dot, not of the word.
Question 25 of 36
Hot spot
Points J, K and L are three vertices of a rectangleJKLM, as shown.
Click the location of the fourth vertex, M.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
In a rectangle, opposite sides are parallel. M must line up vertically with one given vertex and horizontally with another.
Step-by-step solution
Read the vertices: J(-3,\ 2), K(4,\ 2), L(4,\ -3).
J and K share y = 2, so side JK is horizontal (top side). K and L share x = 4, so side KL is vertical (right side).
The missing corner is the bottom-left one. It must be directly below J (same x as J: x = -3) and directly left of L (same y as L: y = -3).
So M = (-3,\ -3).
The fourth vertex is (-3,\ -3).
Check: Side JM goes from (-3, 2) to (-3, -3): vertical, length 5 — same as KL from (4, 2) to (4, -3): length 5 ✓. Side ML is horizontal with length 4 - (-3) = 7, same as JK ✓
Tip: For a rectangle drawn along the grid, the fourth vertex borrows its x from one neighbor and its y from the other.
Question 26 of 36
Fill in
A community garden is shaped like a rectangle. On a coordinate grid measured in yards, its corners are at (-3,\ 2), (4,\ 2), (4,\ -3) and (-3,\ -3).
What is the area of the garden, in square yards?
square yards
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the length of a horizontal side and a vertical side from the coordinates, then use A = \ell w.
Step-by-step solution
Length (horizontal side) from (-3, 2) to (4, 2): same y, so subtract the x-values:
4 - (-3) = 7 \text{ yd}
Width (vertical side) from (4, 2) to (4, -3): same x, so subtract the y-values:
2 - (-3) = 5 \text{ yd}
Area of a rectangle:
A = \ell w = 7 \times 5 = 35
The garden's area is 35 square yards.
Check: Count grid squares: 7 columns across and 5 rows down gives 7 \times 5 = 35 squares ✓
Tip: Distance along a grid line = (bigger coordinate) − (smaller coordinate). Subtracting a negative turns into adding, so 4 - (-3) = 7, not 1. Rectangle area A = \ell w is on the GED formula sheet.
Question 27 of 36
On the coordinate plane below, each grid line represents 2 units.
What are the coordinates of point M?
Hint
Check the numbers on the axes before counting squares. How many units does one square stand for?
Step-by-step solution
Read the scale: the axis labels go 2, 4, 6, \dots, so each grid square is 2 units, not 1.
Horizontal: M is 3 grid lines to the left of the y-axis. 3 \times 2 = 6 units left, so x = -6.
Vertical: M is 2 grid lines above the x-axis. 2 \times 2 = 4 units up, so y = 4.
Point M is (-6,\ 4).
Check: Directly below M the x-axis label is -6, and directly left of M the y-axis label is 4 ✓
Tip: Always read the axis labels first. GED graphs often use scales of 2, 5, 10 or more per grid line.
Question 28 of 36
No calculator
The point (a,\ b) is in Quadrant II.
In which quadrant is the point (b,\ a)?
Hint
First write the signs of a and b. Then remember which position is x and which is y in (b, a).
Step-by-step solution
Quadrant II has the sign pattern (-, +). So a < 0 and b > 0.
In the new point (b, a), the first coordinate is b (positive) and the second is a (negative).
Sign pattern of (b, a): (+, -).
(+, -) means right of the y-axis and below the x-axis: Quadrant IV.
The point (b, a) is in Quadrant IV.
Check: Pick numbers: a = -5, b = 2. Then (a, b) = (-5, 2) is in Quadrant II ✓ and (b, a) = (2, -5) is right and down — Quadrant IV ✓
Tip: Swapping the coordinates of a point reflects it across the line y = x, which sends Quadrant II to Quadrant IV (and IV to II), while I and III stay put.
Question 29 of 36
No calculatorFill in
Point A is at (4,\ -7). It is reflected across the x-axis, and then that image is reflected across the y-axis.
What are the coordinates of the final image?
( , )
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Do one reflection at a time. Each reflection changes the sign of just one coordinate.
Step-by-step solution
Reflect across the x-axis: the y-coordinate changes sign, x stays.
(4, -7) \to (4, 7)
Reflect across the y-axis: the x-coordinate changes sign, y stays.
(4, 7) \to (-4, 7)
Overall, both signs changed: (4, -7) \to (-4, 7).
The final image is (-4, 7).
Check: The original point is in Quadrant IV (+, -). Two reflections across perpendicular axes send it to the opposite quadrant, Quadrant II (-, +), and (-4, 7) is in Quadrant II ✓
Tip: Reflecting across both axes is the same as a half turn about the origin: (x, y) \to (-x, -y). The numbers never swap places.
Question 30 of 36
A parking lot is a rectangle. On a site plan measured in meters, its corners are at (-5,\ 3), (2,\ 3), (2,\ -1) and (-5,\ -1). A fence will go all the way around the lot.
How many meters of fence are needed?
Hint
Find the length of a horizontal side and of a vertical side. Then add all four sides.
Step-by-step solution
Horizontal side from (-5, 3) to (2, 3) (same y):
2 - (-5) = 2 + 5 = 7\text{ m}
Vertical side from (2, 3) to (2, -1) (same x):
3 - (-1) = 3 + 1 = 4\text{ m}
Perimeter = all four sides:
P = 2(7) + 2(4) = 14 + 8 = 22\text{ m}
The fence must be 22 meters long.
Check: Walk around the lot: 7 + 4 + 7 + 4 = 22 ✓ (Area would be 7 \times 4 = 28 square meters — a different question.)
Tip: Fence, border, trim = perimeter (add sides). Carpet, paint, sod = area (multiply). Subtracting a negative coordinate means adding its distance.
Question 31 of 36
No calculator
A city planner's map uses a grid on which each grid line stands for 10 meters.
The new bus stop is marked K. What are its coordinates, in meters?
Hint
Count grid lines from the origin, then multiply each count by the value of one grid line.
Step-by-step solution
Count grid lines from the origin to point K: 3 lines to the left and 2 lines up.
Each grid line is worth 10 units (the labels 20 and 40 on every second line confirm this).
x: 3 lines left = 3 \times 10 = 30 units left, so x = -30.
y: 2 lines up = 2 \times 10 = 20 units up, so y = 20.
Point K is at (-30, 20).
Check:K sits exactly on the horizontal grid line labeled 20 ✓, and halfway between -20 and -40 horizontally, which is -30 ✓
Tip: Before reading any graph, check the scale on both axes. On the GED, grids often count by 2, 5, 10 or more.
Question 32 of 36
Hot spot
Points A(-4,\ -2), B(1,\ -2) and C(3,\ 2) are three vertices of parallelogramABCD (the vertices are listed in order around the shape).
Click the location of vertex D.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
In a parallelogram, the move from B to C is the same as the move from A to D.
Step-by-step solution
In parallelogram ABCD, side \overline{AD} is parallel to and the same length as side \overline{BC}. So the move from A to D equals the move from B to C.
Apply the same move to A(-4, -2):
D = (-4 + 2,\ -2 + 4) = (-2,\ 2)
Vertex D is at (-2, 2).
Check: Side \overline{DC} runs from (-2, 2) to (3, 2): length 5, horizontal. Side \overline{AB} runs from (-4, -2) to (1, -2): length 5, horizontal. Opposite sides are equal and parallel ✓
Tip: A parallelogram need not be a rectangle. Use the same "slide" (right/up amounts) on both pairs of opposite sides instead of assuming right angles.
Question 33 of 36
No calculatorSelect all
The point (a,\ b) is in Quadrant II.
Select all the points below that must lie in Quadrant III.
Select all that apply.
Hint
Write the signs of a and b. Then find the sign of each coordinate: a minus sign in front of a negative number makes it positive.
Step-by-step solution
Quadrant II means a < 0 (negative) and b > 0 (positive).
Then -a > 0 (the opposite of a negative is positive) and -b < 0.
Quadrant III needs the pattern (-, -). Test each point:
(a, -b): (-, -) ✓ Quadrant III.
(-a, b): (+, +) → Quadrant I.
(b, a): (+, -) → Quadrant IV.
(-b, a): (-, -) ✓ Quadrant III.
(-a, -b): (+, -) → Quadrant IV.
The points in Quadrant III are (a, -b) and (-b, a).
Check: Let a = -3, b = 1. Then (a, -b) = (-3, -1) ✓ and (-b, a) = (-1, -3) ✓ are both lower-left; (-a, b) = (3, 1), (b, a) = (1, -3), (-a, -b) = (3, -1) are not ✓
Tip: "-a" is read "the opposite of a", not "negative a". If a is already negative, -a is positive. Plugging in sample numbers is the safest way to check.
Question 34 of 36
Fill in
A hiking trail map is drawn on a grid. Each grid square is 0.5 mile on a side.
A straight path runs along the grid line from the Trailhead to the Lake. How many miles long is the path?
miles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Count the squares between the Trailhead and the Lake first. Then use the scale.
Step-by-step solution
The Trailhead and the Lake are on the same horizontal grid line, so the path is a straight horizontal segment.
Count grid squares along it: the Trailhead is 4 squares left of the vertical axis and the Lake is 3 squares right of it, so the path covers
4 + 3 = 7 \text{ squares}
Each square is 0.5 mile, so
7 \times 0.5 = 3.5 \text{ miles}
The path is 3.5 miles long.
Check: 2 squares make 1 mile, so 7 squares make 7 \div 2 = 3.5 miles ✓
Tip: Count squares (or units) on the grid first, then convert with the scale as the last step. Mixing the two steps is where most errors happen.
Question 35 of 36
Fill in
A triangular flower bed is drawn on a garden plan where each unit is 1 foot. Its corners are at (-3,\ -2), (5,\ -2) and (-3,\ 4).
What is the area of the flower bed, in square feet?
square feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Two sides of this triangle lie along grid lines and meet at a right angle. Use them as the base and height.
Step-by-step solution
(-3, -2) and (5, -2) share y = -2: a horizontal side of length
5 - (-3) = 8 \text{ ft}
(-3, -2) and (-3, 4) share x = -3: a vertical side of length
4 - (-2) = 6 \text{ ft}
A horizontal and a vertical side meet at a right angle at (-3, -2), so they are the base and the height.
Area of a triangle (on the GED formula sheet):
A = \tfrac{1}{2}bh = \tfrac{1}{2}(8)(6) = 24
The flower bed has an area of 24 square feet.
Check: The triangle is half of an 8 \times 6 rectangle, whose area is 48; half of 48 is 24 ✓
Tip: On a coordinate grid, look for a corner where one side is horizontal and another is vertical. Those two sides give the base and height with simple subtraction — no slanted lengths needed.
Question 36 of 36
No calculator
The point (2k - 6,\ k + 1) lies on the y-axis.
What are the coordinates of the point?
Hint
On the y-axis, which coordinate equals 0? Find k, then substitute it into both coordinates.
Step-by-step solution
Every point on the y-axis has x = 0. Set the x-coordinate equal to 0:
2k - 6 = 0
Solve: 2k = 6, so k = 3.
Substitute k = 3 into both coordinates:
x = 2(3) - 6 = 0, \qquad y = 3 + 1 = 4
The point is (0, 4).
The coordinates are (0, 4).
Check:(0, 4) has x = 0, so it is on the y-axis, 4 units above the origin ✓
Tip: The question asks for the point, not for k. After solving for the variable, always reread the question and finish the job.
