Weekly Problem 34

6209 /www/nrich/html/content/id/6209/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image alt="Intersection of square and triangle" height="100" src="icon.jpg" width="100"></mdo:image> The dots are one unit apart. What is the area of the region common to both the triangle and the square (in square units)? If you liked this problem, <a href="http://nrich.maths.org/public/viewer.php?obj_id=1867&part=index">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $$\frac{11}{12} \; \text{square units}$$ <mdo:image alt="solution" height="184" src="solution.jpg" width="200"></mdo:image> We want to find out how far the points $A$ and $B$ (the points on both the triangle and sqaure) are from $Y$. The triangle $OQT$ is $3$ units across by $2$ units down. The triangle $OPA$ is similar to $OQT$, and is half its size (since it is one unit down rather than two). So the point $A$ is $\frac{3}{2}$ units from $O$, so $\frac{3}{2}-1=\frac{1}{2}$ unit from $X$, so $1-\frac{1}{2}=\frac{1}{2}$ unit from $Y$ Similarly, the triangle $BST$ is $\frac{1}{3}$ the size of $OQT$, so the point $B$ is $\frac{2}{3}$ unit from $Z$, so $1-\frac{2}{3}=\frac{1}{3}$ unit from $Y$. So the triangle $AYB$ has area $$\frac{1}{2}\times \frac{1}{2}\times {1}{3} \; \text{square units}= \frac{1}{12}\; \text{square units}$$ so the area of the overlap is $$1-\frac{1}{12}\; \text{square units}=\frac{11}{12}\; \text{square units}$$</mdoxml> 2 4 0 0 1 0 0 Weekly Problem 34 - 2008 What is the area of the region common to this triangle and square? Measures and Mensuration Area Secondary Mapping Document Geometrical reasoning US