Depth · one concept from first principles · step 2 of 4
Box the triangle, subtract the corner triangles
As shown in the figure, find the area of triangle formed by the three points , , and in the coordinate plane.
GeometryNumber system
Your answer
How to solve
Strategy Change Focus / Count the Complement — Change focus from the triangle to what surrounds it. A rectangle with sides parallel to the axes fits snugly around the triangle; the leftover pieces are right triangles whose legs are just differences of coordinates. Subtract them from the rectangle.
1STEP 1
Draw the enclosing rectangle
x runs from -2 to 2 and y from -1 to 1, so the rectangle with corners P(-2, 1), Q(-2, -1), C(2, -1), R(2, 1) encloses the triangle.
P(-2, 1), Q(-2, -1), R(2, 1)
Use the smallest and largest x and y.
6.G.A.3Draw A Diagram2STEP 2
Area of the rectangle
Its width is 2-(-2)=4 and its height is 1-(-1)=2.
PQCR: (2-(-2)) × (1-(-1)) = 4 × 2 = 8
A length along an axis is a difference of coordinates.
7.NS.A.1Identify Subproblems3STEP 3
Areas of the three corner triangles
Read the legs off the grid: APB has legs 3 and 1, BQC has 1 and 4, and ACR has 1 and 2.
△ APB = 1/2 × 3 × 1 = 3/2, △ BQC = 1/2 × 1 × 4 = 2, △ ACR = 1/2 × 1 × 2 = 1
Each corner piece is a right triangle with legs along the grid.
7.G.B.6Identify Subproblems4STEP 4
Subtract
The triangle is what is left of the rectangle after removing the three corner triangles.
8 - (3/2 + 2 + 1) = 8 - 9/2 = 7/2
Whole minus the parts we do not want.
7.G.B.6Change Focus Count The ComplementTriangle is what remains of the rectangle once the three corner triangles are taken away: .
Why?
The rectangle has area .
Why?
Its width is the gap between and , which is , and its height is the gap .
🧱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?
A by rectangle holds rows of unit squares.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Each corner piece is a right triangle with legs along the grid, so its area is half of base times height.
🧱A triangle is half its base times its heightTwo copies of a triangle fit together into a parallelogram on the same base, so one triangle covers half of base times height.
Why?
Taking the three corners out of the rectangle leaves exactly the triangle.
Why?
The triangle and the three corner triangles fill the rectangle with no gaps or overlaps, so their four areas add up to .
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Subtracting the corners undoes adding them, so is the triangle's share.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Answer
7/2
The answer is less than half of the 8-square-unit rectangle, which is expected for a triangle that touches the rectangle's sides at only three points. The shoelace sum 1 × (0-(-1)) + (-2) × (-1-1) + 2 × (1-0) = 1+4+2 = 7, halved, also gives 7/2.
Takeaway
A slanted triangle is a rectangle with three right triangles cut off -- and those are easy.
- Draw the enclosing rectangle
- Area of the rectangle
- Areas of the three corner triangles
- Subtract
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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