Depth · one concept from first principles · step 2 of 4
A square on a given side has two sides to grow
Points and are vertices of square . Find all possible coordinates of the other two vertices, and .
GeometryNumber system
Your answer
How to solve
Strategy Make a Systematic List — Draw the side AB. Because it is vertical, the sides BC and AD must be horizontal and the same length as AB. A horizontal segment can go right or left, so list both cases.
1STEP 1
Find the side length
A and B have the same x-coordinate, so AB is vertical and its length is the difference of the y-coordinates.
AB = 2 - (-1) = 3
Vertical distance is the difference of y-coordinates.
7.NS.A.1Draw A Diagram2STEP 2
Decide the direction of the other sides
BC and AD are horizontal, 3 units long, so C keeps B's y-coordinate -1 and D keeps A's 2, both shifted 3 left or right.
C(-1 ± 3, -1), D(-1 ± 3, 2)
Turning a vertical side by a right angle gives a horizontal side.
7.G.A.2Draw A Diagramand sit units straight to the right of and , or units straight to the left.
Why?
is the vertical gap between and , which is .
🧱Absolute value is distance from zero|a| is how far a sits from 0 on the number line, whichever side it is on, so it is never negative and |a - b| is the gap between a and b.
Why?
Turning side a quarter turn about , one way or the other, lays it along the horizontal line through and keeps its length , so is to the right or to the left of .
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Turning the same way about puts the same units over from , on the same side as .
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
3STEP 3
Case 1: build to the right
Moving 3 to the right from x=-1 gives x=2.
-1 + 3 = 2 → C(2, -1), D(2, 2)
Right means add.
6.NS.C.8Make A Systematic List4STEP 4
Case 2: build to the left
Moving 3 to the left from x=-1 gives x=-4.
-1 - 3 = -4 → C(-4, -1), D(-4, 2)
Left means subtract.
6.NS.C.8Make A Systematic ListAnswer
C(2, -1), D(2, 2) or C(-4, -1), D(-4, 2)
In the first case BC = 2-(-1) = 3, CD = 2-(-1) = 3, and DA = 2-(-1) = 3, all equal to AB with only horizontal and vertical sides, so it is a square. The second case is its mirror image across the line x=-1.
Takeaway
When a shape can face either way, check both sides -- one answer is only half the answer.
- Find the side length
- Decide the direction of the other sides
- Case 1: build to the right
- Case 2: build to the left
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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