Exercises 5 and 6 require the notion of area in the Euclidean plane. Below l've listed theproperties that you will need.Properties of Area. Area is a function that assigns a positive real mumber to (a class of)subsets of the Euclidean plane. It has the following properties:(1) The area of a rectangle of side lengths a and b is ab (a rectangle is a quadrilateral inwhich all interior angles are right).(2) Given a finite collection of sets E1,..., Ex in the plane that are pairwise disjoint (thatis, E,nE, = Ø for distinet i, j e {1,...,k}), the area of Ej UE2U.U E is equal tothe sum of the individual areas, that is, Area(E U E2U...U ER) = E-1 Area(Eg).(3) The area of any line segment is zero.(4) Congruent triangles have the same area.Definition. A triangle is right if it contains a right angle. The side of a right triangleopposite the right angle is called the hypotenuse; the other sides are called legs.Exercise 5. Let AABC be a triangle, let D be the point on the (extended) line AB suchthat DC 1 AB, and let b = AB and h = DC. Prove that the area of AABC is bh. (Hint:first, prove that the area of a right triangle is kab, where a and b are the lengths of the legsof the right triangle.)Figure 3: A helpful figure for proving the Pythagorean theorem.Exercise 6. Prove the Pythagorean Theorem, that is, given a right-triangle with hy-potemuse of length e and legs of length a and b, prove that a? +b = 2.

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