1. Right Angled TrianglesWorking with Pythagorean Theorem Stonemasonry Department 2011
2. Types of Triangles Isosceles Right Angled An isosceles triangle has The right angled two sides of the same triangle consists of one length and two angles angle of 90° and a side of the same measure. opposite the right All angles add to 180° angle known as the hypotenuse. Scalene Equilateral A scalene triangles has An equilateral triangle three sides of different has three sides of the lengths and three same length and three angles of different sizes. identical angles (60°).
3. Right Angled TrianglesThe angles are The long diagonalmarked with arcs. If side of a right angledtwo angles have the triangle is known assame number of arcs the hypotenuse.it means they areboth the same. All right angled triangles have an The marks on each side of the angle which measure triangle indicate that each side is 90°. of different lengths. If two sides had the same markings it would mean they share the same length.
4. The Three, Four, Five Rule a = 5m b = 3m c = 4m The 3,4,5 rule states that if the sides of a right 25m² angled triangle are 3m and 4m then the hypotenuse would measure 5m. 9m² This comes from a theory called pythagoras theorem which states that: a² = b² + c² 16m² Where a, b and c are the names given to the sides of the right angled triangle
5. Pythagorean Theory However, not all triangles have sides whichmeasure 3m, 4m and 5m so how do we check if they are right angled triangles? 10 x 10 = 100m² We can still use pythagorean theory which states that: 36m² 6x6= a² = b² + c² Lets look at the triangle below: 64m² 8x8= b = 6m a = 10m 36m² + 64m² = 100m² This means that the triangle is a right angled triangle. c = 8m
6. Pythagorean Theory The theory works with all right angled triangles a² = b² + c² 225m² Lets look at the triangle below: 81m² a = 15m 144m² b = 9m 81m² + 144m² = 225m² c = 12m This means that the triangle is a right angled triangle.
7. Pythagorean Theory a² = b² + c² Lets look at the triangle below: 81m² 64m² a = 9m 25m² b = 8m 64m² + 25m² = 89m² c = 5m This means that the triangle is a NOT a right angled triangle.
8. Determining Right Angles Determine whether the triangles shown below are right angled. a² = b² + c² a = 25m a = 0.5m b = 15m b = 0.3m c = 20m c = 0.4m a = 89m a = 9m b = 5m b = 8m c = 8m c = 5m
9. Finding the Hypotenuse We can also use the theory to calculate thelength of the hypotenuse provided we know the triangle is a right angled triangle. ? a² = b² + c² Lets look at the triangle below: 16m² 64m² a = ?m b = 4m 16m² + 64m² = 80m² We then find the square root (√)of 82 to c = 8m determine the length of the hypotenuse: √80 = 8.94m
10. Finding the Hypotenuse Determine the length of the hypotenuse for each of the triangles shown below using pythagoras theory. a² = b² + c² a = ?m a = ?m b = 6m b = 0.4m a=9.2m a=0.8m c = 7m c = 0.7m a = ?m a = ?m b = 2m b = 8.3m a=3.6m a=12.8m c = 3m c = 9.7m
11. Finding the Length of One Side We can also use the theory to calculate the length of one side provided we know the triangle is a right angled triangle. 100m² a² = b² + c² Lets look at the triangle below: ? 64m² a = 10m b = ?m ? + 64m² = 100m² So we subtract 64 from 100 to get 36 and c = 8m then find the square root (√)of 36 to determine the length of the side: √36 = 6m
12. Finding the Length of One Side Determine the length of the missing side for each of the triangles shown below using pythagoras theory. a² = b² + c² a = 18m a = 8.2mb = 11.3m b = 3.2m c=14m c=7.6m c = ?m c = ?m a = 3m a = 12.5m b = 1m b = ?m c=2.8m b=7.9m c = ?m c = 9.7m
13. Image References The image on the title slide of this presentation was sourced from Wikipedia at: http://en.wikipedia.org/wiki/File:Kapitolinischer_Pythagora s_adjusted.jpg This image was made available under creative commons
14. Developed by The Stonemasonry Department City of Glasgow College 2011