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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Congruence

Congruence is a defining factor between two or more shapes. For two shapes to be congruent, they must meet the following criteria: 

  • Have an equal number of sides
  • The corresponding side lengths in each shape are equal 
  • The corresponding angles in each shape are equal

Consider Figures 1 and 2:

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We see in Figure 1 that the two rectangles have side lengths of 2cm and 4cm, and four 90° angles. Since the rectangles meet all of the criteria, we can conclude that they are congruent

In Figure 2, one rectangle has side lengths of 2cm and 4cm and the other has side lengths of 1cm and 3cm. Although they both have four 90° angles, they do not pass the second criteria and so they are not congruent.

What can we do to a shape so that the result is congruent to the original?

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Similarity

Similarity in geometry is used to describe two shapes whose corresponding angles are equal, and side lengths are in proportion. Shapes that are congruent are also similar, but shapes that are similar are not always congruent. 

Consider Figures 3 and 4: 

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Figure 3 shows two triangles, one bigger than the other. Looking closely, it is visible that they have the same corresponding angles and side lengths that are in proportion (dividing the lengths of the first triangle by approximately 1.9 yields the lengths of the second triangle). 

In Figure 4, we can see that the second triangle fits perfectly into the first triangle and can be resized/scaled to become congruent to it. 

Since the two triangles meet the necessary criteria, they are similar triangles. 

Example

Mark is given a right-angled triangle with one side and the hypotenuse having lengths 52 and 65, respectively. His teacher asks him to scale the triangle so that the hypotenuse has length 5. What are the other two side lengths of the scaled triangle? 

See the video below for the solution: 

Real-life Applications

How can we apply the concept of similarity to the workplace?

The shape of triangles is present in the construction of almost every building, bridge or other form of architecture. Physics can be used to explain why they are so strong, but similarity is the medium by which triangles become present.

Some examples of real-life applications are: 

  • determining the height of an object
  • scale models of buildings, scaling in construction, etc.
  • analysis of bridge stability 

Example

Anastasia is tasked with determining the height of a pine tree on the college campus. She aligns the tip of her shadow with the tip of the tree's shadow. If Anastasia is 150 centimeters tall, her shadow is 2 meters long and the tree's shadow is 6 meters long, how tall is the tree? 

See the video below for the solution:

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