Triangles’ Theorems.pptx

triangles theorem

1. Triangles’ Theorems. Chem. Eng. Daniel Esparza Rivera

2. Triangle. Shape formed of 3-line segments that intersect 2 to 2 at three points. The intersection points are called vertices. The line segments are called sides. Each pair of sides formed an internal angle. Thus, a triangle is formed of 3 sided and 3 internal angles.

3. Classification By their sides. By their angles Acute triangle Right triangle Obtuse triangle Equilateral triangle Isosceles triangle Scalene triangle

4. Properties. The sum of their internal angles is always 180°. The sum of the length of 2 sides of a triangle is always bigger than the length of the other side.  

5. Congruent and similar triangles. Congruent triangles. Similar triangles.  

6. 1.- Leticia's cat has climbed on top of a post. Leticia can see its cat reflected in a puddle. If her eyes' height is 1.44 m, how tall is the post? A’ B’ C’ A C B According to similar triangles theorem we have that   We have values for A’, B and B’ A’= 1.44 m. A=? B’= 1.6 m. B= 4 m. From the formulas we can substitute and solve…      

7. Postulates on similar triangles. The ratio of the perimeters of two similar triangles is equal to the ratio of any pair of corresponding sides. In a right triangle, the height perpendicular to the hypothenuse form two triangles that are similar to each other and similar also to the biggest triangle  

8. Thales's theorem. If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.  

9. Example. A person that height 6 ft is stand at 8 ft from a lamp post. The shadow of the person projected over the ground is of 12 ft long. What is the height of the lamp post? From Thales theorem:   A B C A’ C’ Data:   Substituting and solving:  

10. Pythagoras’ Theorem Explains the relation between the three sides of a right-angled triangle. A B C The Pythagoras’ Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.   hypotenuse

11. Proof of the Pythagoras’ Theorem. By the postulates of similar triangles, we have that If we use the formulas and relate with we can have Now relating with we can obtain If we sum both expressions, we obtain  

12. What is the length of the ladder? It's 6 ft. from the house at the bottom and touches the wall 14 ft. up at the top.   a b c

13. Theorems are of very importance and significance since they help us not only to solve math problems, but also to develop a better and deeper understanding of the concepts. Theorems as the Thales’ and Pythagoras’ can help to solve problems in the day by day, and they are used in many areas such as architecture, construction, navigation, etc. Examples of the application in real life of the Pythagoras' Theorem. https://www.youtube.com/watch?v=gRf780Pce7o