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Exercise 5. Let AABC be a triangle, let D be the point on the (extended) line AB such that DC 1 AB, and let b = AB and h = DC. Prove that the area of AABC is Įbh. (Hint: first, prove that the area of a right triangle is jab, where a and b are the lengths of the legs of 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 a² +b =2.

Exercise 5. Let AABC be a triangle, let D be the point on the (extended) line AB such that DC 1 AB, and let b = AB and h = DC. Prove that the area of AABC is Įbh. (Hint: first, prove that the area of a right triangle is jab, where a and b are the lengths of the legs of 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 a² +b =2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercises 5 and 6 require the notion of area in the Euclidean plane. Below l've listed the properties 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 in which all interior angles are right). (2) Given a finite collection of sets E1,..., Ex in the plane that are pairwise disjoint (that is, E,nE, = Ø for distinet i, j e {1,...,k}), the area of Ej UE2U.U E is equal to the 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 triangle opposite 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 such that 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 legs of 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.
Transcribed Image Text:Exercises 5 and 6 require the notion of area in the Euclidean plane. Below l've listed the properties 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 in which all interior angles are right). (2) Given a finite collection of sets E1,..., Ex in the plane that are pairwise disjoint (that is, E,nE, = Ø for distinet i, j e {1,...,k}), the area of Ej UE2U.U E is equal to the 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 triangle opposite 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 such that 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 legs of 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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