Pythagoras’ Theorem

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Pythagoras’ Theorem by Mind Map: Pythagoras’ Theorem


2. Proof for Pythagoras

2.1. It is a big square, with each side having a length of a+b, so the total area is: A = (a+b)(a+b) Now let's add up the areas of all the smaller pieces: First, the smaller (tilted) square has an area of A = c2 And there are four triangles, each one has an area of A=½ab So all four of them combined is A = 4(½ab) = 2ab So, adding up the tilted square and the 4 triangles gives: A = c2+2ab The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as: (a+b)(a+b) = c2+2ab NOW, let us rearrange this to see if we can get the pythagoras theorem: Start with: (a+b)(a+b) = c2 + 2ab Expand (a+b)(a+b): a2 + 2ab + b2 = c2 + 2ab Subtract "2ab" from both sides: a2 + b2 = c2

2.2. Now we start with four copies of the same triangle. Three of these have been rotated 90o, 180o, and 270o, respectively. Each has area ab/2. Let's put them together without additional rotations so that they form a square with side c. The square has a square hole with the side (a-b). Summing up its area (a-b)2 and 2ab, the area of the four triangles (4·ab/2), we get c2 = (a-b)2+2ab = a2-2ab+b2+2ab = a2+b2

3. What is Pythagoras?

3.1. The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypoteneuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2.

3.2. h is the side of the larger square, and the hypotenuse of the triangle

4. Converse of Pythagoras Theorem

4.1. If the squared of one side is equal to the sum of squared of the other two sides, the triangle is a right-angled triangle

4.2. PROOF

5. Applications of the theorum

6. Toa Cah Soh

6.1. Toa

6.1.1. When finding an angle: 1. take the opposite over the adjacent and equate it to the angle that you want to find multiplied by the tangent sign (note: tangent angle can NEVER be a right angle) 2. throw the tangent over the equal gin and multiply (opp/adj) by the negative tangent 3. punch it in the calculator to get the angle

6.1.2. When finding the length of a side: 1. tangent angle = opp/given adj or given opp/adj 2. if finding for opp, take the tangent angle multiplied by the given adj/ if finding adj, take the given opp and divide it by the tangent angle

6.2. Cah

6.2.1. When finding an angle: 1. cosine the angle you are looking for = given adj/given hypo 2. throw the cosine over the equal sign 3. multiple adj/hypo by negative cosine

6.2.2. When finding length: 1. cosine given angle = adj/given hypo or given adj/hypo 2a. finding adj: multiply cosine angle by given hypo 2b. finding hypo: divide cosine angle by given adj

6.3. Soh

6.3.1. When finding an angle: 1. sine the angle you are looking for = given adj/given hypo 2. throw the sine over the equal sign 3. multiple adj/hypo by negative sine

6.3.2. When finding length: 1. sine given angle = opp/given hypo or given opp/hypo 2a. finding opp: multiply sine angle by given hypo 2b. finding hypo: divide sine angle by given opp

7. If the squared of one side of the triangle is more than the sum of the squared of the other two sides, the triangle is obtuse.

8. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras.

9. History of Pythagoras

9.1. takes its name from the ancient Greek mathematician Pythagoras (569 B.C.-500 B.C.)

9.2. B.C.Ancient clay tablets from Babylonia indicate that the Babylonians in the second millennium B.C., 1000 years before Pythagoras, had rules for generating Pythagorean triples, and understood the relationship between the sides of a right triangle, and, in solving for the hypotenuse of an isosceles right triangle, came up with an approximation of accurate to five decimal places.

9.2.1. A Chinese astronomical and mathematical treatise called Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gives a statement of and geometrical demonstration of the Pythagorean theorem.

9.3. The concept was not original as Babylonians had used it was earlier in around 1800

9.4. India: Baudhayana Sulba Sutra, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. The Apastamba Sulba Sutra (c. 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation.

9.5. video

10. Corollary of Pythagoras Theorem

10.1. If the squared of one side of the triangle is less than the sum of squared of the other two sides, the triangle is acute.

11. Trigo Ratios

11.1. Conditions

11.1.1. Trigo Ratios can only be used on RIGHT ANGLED triangles

11.2. Identifying Lengths

11.2.1. The Opposite length (Opp) is the length opposite the angle you are finding

11.2.2. The Hypotenuse (Hyp) is the longest side of the right angled triangle

11.2.3. The Adjacent length is the length next to the angle you are finding

11.2.4. eg. angle BAC: opp-a, hyp-c, adj-b

11.3. Finding Trigo ratios

11.3.1. TOA : tan-1 (opp/adj)

11.3.2. CAH : cos-1 (adj/hyp)

11.3.3. SOH : sin-1 (opp/hyp)

11.4. Application

11.4.1. used for finding the height of buildings, mountains, etc.

11.4.2. used in navigation, to find the distance of the shore from a point in the sea

11.4.3. used in oceanography to find the height of the tides in the sea

11.4.4. used in finding the distance between celestial bodies

11.4.5. Architects use trigonometry to calculate structural load, roof slopes, ground surfaces, etc.

12. Results from Clinometer Task

12.1. 1 kf is approximately 22cm

12.2. Lamp outside Sports Hall

12.2.1. 49 degrees

12.2.2. 8 kerifeet

12.2.3. height of lamppost = tan(49) (8) = 9.20 kf (kerifeet) = 2.02m

12.3. small netball hoop

12.3.1. 33 degrees

12.3.2. 7 kerifeet

12.3.3. Height of small netball hoop = tan (33) (7) = 4.55kf = 1.00m

12.4. fan in Sports hall

12.4.1. 26 degrees

12.4.2. 10 kerifeet

12.4.3. Height of fan = tan (26) (10) = 4.88 kf = 1.07m

12.5. timing/score board in Sports hall

12.5.1. 41.5 degrees

12.5.2. 17-18 kerifeet

12.5.3. Height of scoreboard = tan (41.5) (17.5) = 15.48kf = 3.41m