Here's an interesting challenge:
a. What percentage of the area of \(\square ABCD\) is occupied by \(\triangle AEF\)?
b. Prove that \(\triangle COE\) is isosceles
Not too easy, not too difficult.
a. What percentage of the area of ABCD is occupied by \(\triangle\)EFA?
Hi Mathhemathh!
\(\overline{EF}=\overline{CD}\cdot\sqrt{2}-\overline{CD}\\ \overline{EF}=10\cdot\sqrt{2}-10\\ \)
\(A_\triangle=\frac{\overline{EF}}{2}\cdot (\overline{CD}\cdot \sqrt{2}-\frac{\overline{EF}}{2}) \)
\(A_\triangle=\frac{10\cdot\sqrt{2}-10}{2}\cdot (10\cdot\sqrt{2}-\frac{10\cdot\sqrt{2}-10}{2})\\ A_\triangle=(10\cdot\sqrt{2}-10)^2\)
\(A_\triangle=17.15729\)
\(A_\square : A_\triangle=100\% :x\\ x=\frac{ A_\triangle\cdot 100\%}{A_\square}\\ x=\frac{17.15729\cdot 100\%}{100}\\ \color{blue}x=17.15729\%\)
\(a:\ \triangle EFA\ has\ 17.15729\%\ of\ \square ABCD \)
!
Let the center of the circle with a radius of 5 be centered at (0,0) and let A = (-5,5)
The midpoint of EF is (5 cos (45°), -5sin(45°) ) = ( 5/√2, -5/√2)
Let this midpoint be = G
So the height of triangle AEF = GA = √ [( 5/√2 + 5) ^2+ ( 5 +5/√2)^2 ] = 5 + 5√2 units =
5 ( 1 + √2) units = 5 (√2 + 1) units
Let H be the point of tangency where BD touches the circle
And FG = FH since these are tangents drawn to a circle from the same exterior point, F
And angle HOF = 22.5° and angle OFH = 67.5°...so...using the Law of Sines
FH / sin (22.5°) = OF / sin (67.5°)
FH = 5 sin (22.5°) / sin (67.5°)
FH = 5 sin (22.5°) / cos (22.5°)
FH = 5 * tan (22.5°) = 5 * [ 1 - cos (45°) ] /[ sin (45°) ] = 5 * [ 2 - √2] / [ √2] units =
5 (√2 -1) = FG
So...the area of AEF = (GA) ( FG) = 5 (√2 + 1) * 5 ( √2 -1) units^2 =
25 (2 - 1) =
25 units^2
The area of square ABCD =10*10 = 100 units^2
So area ΔAEF = 25 / 100 = 25% of square ABCD
Here's a pic :