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Pythagorean inequalities assist in determining whether a triangle is a right acute or obtuse triangle.

Pythagorean Inequalities:

In ΔABC “c” is the length of the longest side.

If a² + b² > c², then ΔABC is an acute triangle

If a² + b² < c², then ΔABC is an obtuse triangle

The Pythagorean Theorem always works for right triangles. It is also used to classify triangles that are not right triangles.

Example: Let a = the smallest side
Let b = the middle length side
Let c = the longest side

Triangle 1 – An Acute Triangle

Image shows a triangle with side lengths 9 centimeters, 8 centimeters and 7 centimeters.

Fill in the Blanks

The Pythagorean Theorem is ______?______
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a² + b² = c²
The shortest side "a" is: __?___
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7 cm
The middle side "b" is: ___?___
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8 cm
Square "a" and "b" and add them together: ____?____
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49 + 64 = 113
The longest side "c" is: ___?____
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9 cm
Square "c": ___?____
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9² = 81
a² + b²_?__c² ?
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>
When a² + b² > c² the triangle is an ___?___ triangle.
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acute

Triangle 2 – An Obtuse Triangle

Image shows a triangle with side lengths 8 centimeters, 16 centimeters and 25 centimeters.

The Pythagorean Theorem is ____?______
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a² + b² = c²
The shortest side "a" is _____?_____
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16 cm
The middle side "b" is ____?______
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19 cm
Square "a" and "b" and add them together. ____?_____
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256 + 361 = 617
The longest side "c" is: ____?_____
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25 cm
Square "c": _____?____
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25² = 625
a² + b² _?_ c² ?
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<
When a² + b² = c² the triangle is an ____?_____ triangle.
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obtuse

If a² + b² < c²,
then the triangle is an ____?_____ triangle. Interactive popup. Assistance may be required.

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obtuse

If a² + b² > c²,
then the triangle is an ____?____ triangle.
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acute

You try it!

The measurements of 3 sides of a triangle are given. Identify the following as a right triangle, acute triangle, or obtuse triangle.

The measurements of 3 sides of a triangle:

5, 12, 13 ___?____
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Right Triangle
7, 8, 9 ____?____
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Acute Triangle
6, 8, 10 ___?____
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Right Triangle
13, 15, 21 __?___
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Obtuse Triangle
7, 10, 12 ___?___
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Acute Triangle