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.

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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