Geometry

Congruent and Similar Triangles

Congruent and Similar Triangles: Level 3 Challenges

         

ABCDABCD is a square with AB=13AB=13. Points EE and FF are exterior to ABCDABCD such that BE=DF=5BE=DF=5 and AE=CF=12AE=CF=12.

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

The area of the largest square which can be inscribed in a triangle with side lengths 3,4,53, 4, 5 is ab\dfrac{a}{b}.

Find a+ba+b, where aa and bb are coprime, positive integers.

BCDFBCDF is a rectangle. Triangle ABEABE has an area of 2 cm2 2 \text{ cm}^2 . Triangle BEFBEF has an area of 3 cm2 3 \text{ cm}^2 .

Find the area of the blue region (in cm2 \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 ABCABC is similar to triangle BFA,BFA, then find ABBC.\dfrac{AB}{BC}.

There is a common tangent which intersects the three tangent circles shown above at E,E, F,F, and G.G. If EF=6 EF = 6 and FG=3 FG = 3 , find the area of the orange region. Use the approximation π227.\pi \approx \frac{22}{7}.

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