Problem 14 · Two triangles in the square
They share the overlap: the difference equals that of the full triangles.
Integer answer, at most 4 digitsThe figure shows a square and two shaded triangles inside it. What is the difference between the areas of the two shaded triangles?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: the two shaded regions are two big triangles minus the same overlap quadrilateral: subtracting, the overlap cancels.
Square of side $16$ with origin at the bottom-left vertex. Tall triangle: vertices $(8,16)$, $(12,0)$ and $(16,2)$; low triangle: $(0,6)$, $(12,0)$ and $(16,2)$. By the determinant formula:
$$A_1 = \tfrac{1}{2}\left|4 \cdot (-14) - (-16) \cdot 8\right| = 36, \qquad A_2 = \tfrac{1}{2}\left|12 \cdot (-4) - (-6) \cdot 16\right| = 24.$$
$$\Delta = (A_1 - \text{overlap}) - (A_2 - \text{overlap}) = 36 - 24 = 12\ \text{cm}^2.$$
Answer: 12