Monday, January 18, 2016

Division M2 session on Jan 17, 2016

Instructor:Ahmed Hefny
Handout:Level 5 Geometry Problems (All homework except problem 5). Please download the handout by clicking here.
Problems Solved in Class:Problem 5
Concepts:Pythagorean theorem, congruent triangles, similar triangles
Student Difficulties:Square roots, solving geometry problems by constructing intermediate shapes.
Notes:Congruent Triangles
Two triangles are congruent if they have equal side lengths and angle measures. 
We do not need to test ALL lengths and measures; there are minimal tests for congruency.  

Usually the solution strategy goes as follows: 
  • There is missing piece of information about triangle A (e.g. missing side length) 
  • The information we have about A is sufficient to establish congruency with another triangle B using one of the minimal tests. 
  • Once congruency is established, we determine missing information about A from what we know about B. 

Minimal tests for congruency: 
  • Three sides 
  • Two sides and the angle between them 
  • Two angles and the side between them 
  • If the two triangles are right. Any two sides suffice (due to Pythagorean theorem) 

How do we know these tests are sufficient ? 
The information used in the tests uniquely identify all lengths and angles of a triangle. 

Example 1: 
Knowing the lengths of all sides uniquely determines the shape of a triangle. Figure 1 shows how to draw a triangle knowing the lengths of the three sides using a ruler and a compass. 

Example 2: 
Knowing the lengths of two sides and the measurement of an angle other than the one between them does not uniquely determine the shape of a triangle. 

Similar triangles: 
  • The ratios of side lengths are the same for all corresponding sides. 
  • Corresponding angles have equal measures. 
Each property implies the other one. 

If two shapes are similar with side length ratio r, the are ratio is r*r and the volume ratio is r*r*r. 

Important right triangles: 

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