3-4-5 Triangle

Problem
From the right triangle ABC shown below, AB = 40 cm and BC = 30 cm. Points E and F are projections of point D from hypotenuse AC to the perpendicular legs AB and BC, respectively. How far is D from AB so that length EF is minimal?
 

Problem
From the figure shown, AB = diameter of circle O₁ = 30 cm, BC = diameter of circle O₂ = 40 cm, and AC = diameter of circle O₃ = 50 cm. Find the shaded areas A₁, A₂, A₃, and A₄ and check that A₁ + A₂ + A₃ = A₄ as stated in the previous problem.