We see the triangle is a right triangle, so \(r^2+(2r)^2=10, r^2+4r^2=10, 5r^2=10, r^2=2, r=\sqrt2, \sqrt2\times 2\sqrt2=4, \frac{4}{2}=2\)
The area of the triangle is 2.
Use the Pythagorean theorem.
\(x^2+9x^2=5, 10x^2=5, x^2=\frac{1}{2}, x=\frac{\sqrt2}{2}\)
Then \(\frac{\sqrt2}{2}\times \frac{3\sqrt2}{2}=\frac{3}{2}, \frac{3}{2}\times \frac{1}{2}=\frac{3}{4}\)
The area of the triangle is 0.75.
\(\Delta ABE, \Delta ECD\) are similar.
\(\frac{BE}{CD}=2\), so \(CD=\frac{BE}{2}=\frac{EC}{2}\).
Using Pythagorean theorem, \(AD=\sqrt5DE\) , \(DE=BE\sqrt5/2\)
\(\frac{AD}{BC}=5DE/4DE\)
5/4
#4: I think there is insufficient explanation here.
You are very welcome!
:P
