Lesson 1: Pythagorean Theorem - Determining if a Triangle has a Right Angle

   Constructing Knowledge

Sometimes a triangle appears to contain a right angle (90° angle) even when it doesn't. Diagrams can be misleading, and even protractors cannot be used for confirmation if the diagram is not drawn to scale. Unless the angle is marked as a right angle, or the context tells you it is, you cannot assume it is one.

It is the relationship between the side lengths of a triangle that dictates whether a triangle is a right triangle. If the sum of the squares of the legs (shortest two sides), a2 + b2, is equal to the square of the longest side, c2, the triangle is a right triangle. If a2 + b2 ≠ c2, the triangle is not a right triangle.

   Multimedia

A video describing how to determine if a triangle is a right triangle is provided.


EXAMPLE 1

Is the triangle shown a right triangle?



Solution


Step 1: Label the sides of the triangle.



Step 2: Calculate the square of the longest side (one side of the equation a2 + b2 = c2).

The longest side, c, is 13 units long

c2 = 132
= 169 square units

Step 3: Calculate the sum of the squares of the other two sides (one side of the equation a2 + b2 = c2).

The other two sides, a and b are 12 and 5

a2 + b2 = 122 + 52
= 144 + 25
= 169 square units

Because both sides of the equation are 169 square units, this triangle is a right triangle.

Alternate Solution


You can also do these calculations in one step.

a2 + b2 = c2
122 + b2 = c2
144 + 25 = 169
169 = 169

Because both sides of the equation are 169 square units, this triangle is a right triangle.


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