Try this beautiful problem from Geometry: Area of a Triangle
The area of rectangle \(ABCD\) is \(72\) units squared. If point \(A\) and the midpoints of \(BC\) and \(CD\) are joined to form a triangle, the area of that triangle is
Geometry
Triangle
square
Answer: \(27\)
AMC-8 (2000) Problem 25
Pre College Mathematics
Area of the triangle =\(\frac{1}{2} \times base \times height \)
Can you now finish the problem ..........
Therefore area of the shaded region i.e area of the \(\triangle AEF\)=area of the square- area of \((\triangle ADE +\triangle EFC +\triangle ABF)\)
can you finish the problem........
Given that area of the rectangle ABCD=72
Let length AB=\(x\) and length of CD=\(y\)
Therefore DE=EC=\(\frac{y}{2}\) and BF=FC=\(\frac{x}{2}\)
Area of ABCD=\(xy\)=72
Area of the \(\triangle ADE=\frac{1}{2}\times DE \times AD= \frac{1}{2}\times \frac{x}{2} \times y =\frac{xy}{4}=\frac{72}{4}=18\)
Area of the \(\triangle EFC=\frac{1}{2}\times EC \times FC= \frac{1}{2}\times \frac{x}{2} \times \frac{y}{2} =\frac{xy}{8}=\frac{72}{8}=9\)
Area of the \(\triangle ABF=\frac{1}{2}\times AB \times BF= \frac{1}{2}\times y\times \frac{x}{2}=\frac{xy}{4}=\frac{72}{4}=18\)
Therefore area of the shaded region i.e area of the \(\triangle AEF\)=area of the square- area of \((\triangle ADE +\triangle EFC +\triangle ABF)=72-(18+8+18)=27\)
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