Congruent and Similar Triangles

Congruent and Similar Triangles: Level 3 Challenges


\(ABCD\) is a square with \(AB=13\). Points \(E\) and \(F\) are exterior to \(ABCD\) such that \(BE=DF=5\) and \(AE=CF=12\).

If the length of \(EF\) can be represented as \(a\sqrt b, \) where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of any prime, then find \(ab\).

The area of the largest square which can be inscribed in a triangle with side lengths \(3, 4, 5\) is \(\dfrac{a}{b}\).

Find \(a+b\), where \(a\) and \(b\) are coprime, positive integers.

\(BCDF\) is a rectangle. Triangle \(ABE \) has an area of \( 2 \text{ cm}^2 \). Triangle \(BEF \) has an area of \( 3 \text{ cm}^2 \).

Find the area of the blue region (in \( \text{cm} ^2 \)). Give your answer to 1 decimal place. (The figure is not drawn to scale. )

Note: This question is supposed to be harder than it looks like. Do not assume anything that is not stated in the problem! (Hint: Angle E is NOT 90 degrees, nor is A the midpoint of BC.)

If triangle \(ABC\) is similar to triangle \(BFA,\) then find \(\dfrac{AB}{BC}.\)

There is a common tangent which intersects the three tangent circles shown above at \(E,\) \(F,\) and \(G.\) If \( EF = 6 \) and \( FG = 3 \), find the area of the orange region. Use the approximation \(\pi \approx \frac{22}{7}.\)


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