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Pythagoras by Mind Map: Pythagoras

1. But from the supposition, a^2 + b^2 = c^2 (2). right triangle 3. From (1) and (2) x^2 = c^2. Since x and c are both positive (Can you see why?), we can therefore conclude that x = c. This means that length of the three corresponding pairs of sides of triangle ABC and triangle DEF are equal. Therefore, by SSS Congruence, \triangle ABC \cong \triangle DEF. Since F and C are corresponding angles, \angle F = \angle C = 90 degrees. And hence we have proved that triangle ABC is right angled at C

2. Sides of a triangle

3. real life application

3.1. In construction, this theorem is one of the methods builders use to lay the foundation for the corners of a building.

3.2. It is used in computing surface areas, volumes and perimeters of different geometric shapes, converting between polar and rectangular coordinates and computing the distance between particular points on a plane. The distance formula is one of its most frequent applications.

3.3. Painting a wall: to use to calculate the height of a ladder needed to safely place the ladder against the wall so it won't fall. In this case, the ladder is literally the hypotenuse and we can calculate the length using the measurements we have.

3.4. architects and engineers use this formula extensively when building ramps:

3.4.1. The owners of a house want to convert a stairway leading from the ground to their back porch into a ramp. The porch is 3 feet off the ground, and due to building regulations, the ramp must start 12 feet away from the base of the porch. How long will the ramp be?

3.4.2. To solve a problem like this one, it often makes sense to draw a simple diagram showing the legs and hypotenuse of the triangle.

3.4.3. :U07_L2_T1_tt_img6.png

3.4.4. Looking at the diagram, we can identify the legs and the hypotenuse of the triangle in the problem we need to solve. We know that the triangle is a right triangle since the ground and the raised portion of the porch are perpendicular—this means we can use the Pythagorean Theorem to solve this problem. We are given the lengths of legs a and b, so we can use that information to find the length of c, the hypotenuse.

3.4.5. Example

3.4.6. Problem

3.4.7. Find c when a = 3 and b =12

3.4.8. Pythagorean Theorem

3.4.9. Substitute known values in for a and b.

3.4.10. Simplify

3.4.11. Combine like terms =

3.4.12. Take the square root of both sides

3.4.13. Answer

3.4.14. 12.37 ≈ c

3.4.15. The ramp will be just about 12.37 feet long.

4. Converse of Pythagoras' Theorem

4.1. If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle

4.2. To find out whether a triangle is a right -angled triangle or not

4.3. Let DEF be a triangle such that EF = a, DF = b and right angled at F (see figure below). If we let DE = x, since DEF is a right triangle, by the Pythagorean Theorem a^2 + b^2 = x^2 (1).

5. Who was Pythagoras?

5.1. He travelled around the world (Syria, Italy, Chaldea, Egypt) learning under great philosophers

5.2. He was a Greek mathematician and philosopher who lived from 570-495 BC

5.3. Pythagoras was born in the eastern Aegean island of Samos, Greece in 570 BC

5.4. He noticed that vibrating strings produce harmonious tones only when the ratios between the lengths of the strings are whole numbers. He later realized that these ratios could be extended to other instruments.

5.5. He was probably most well known for his contributions to Mathematics and especially geometry

5.6. He believed that everything could be reduced to numbers and these numbers had their own characteristics, strengths and weaknesses.

6. Proof

6.1. Suppose we are given any right triangle with sides of lengths A, B, C. In order to show that A2 + B2 = C2.

6.2. A2 + B2 = C2

6.3. it is enough to show for any set of three similar figures whose widths relate to each other in the proportions A:B:C, that the area of the largest figure is the sum of the two smaller figures. This follows because area scales like the square of the width. The figures can be any shape.

6.4. So, let's use the original triangle for the shape of the figures that we'll use! Figure 1 shows that we've placed 3 blue triangles along the sides of the ABC right triangle so that their widths clearly relate to each other in the proportions A:B:C.

6.5. The two smaller blue triangles are congruent (identical in form) to the light and dark green right triangles, and the largest blue triangle is congruent to the entire ABC triangle. Since the sum of the areas of the green triangles equals the area of the ABC triangle, the sum of the areas of the smaller blue triangles equals the area of the largest blue triangle, as was to be shown!

7. Pythagoras theorem

7.1. The Pythagoras theorem argues that the square of the hypotenuse (the side of the triangle opposite the right angle) is equivalent to the sum of the squares on the other two sides

7.2. a^2 + b^2 = c^2 when c is the hypotenuse

8. Trigonometric ratios

8.1. Ratios (abbreviations)

8.1.1. cos = adj/hyp

8.1.2. tan = opp/adj

8.1.3. sin = opp/hyp

8.1.4. TOA CAH SOH

8.2. Application