Question

topic: radicals and pythagorean theorem
learning goal: i will be able to plug values into the pythagorean theorem to determine if a triangle is a right triangle.
proving if a triangle is a right triangle:

1. label the side lengths a, b, and c (remember c has to be the longest side)
2. plug into the appropriate space in the formula
3. if ( a^{2}+b^{2}=c^{2} ), the triangle is a right triangle

if ( a^{2}+b^{2}
eq c^{2} ), the triangle is not a right triangle
examples:

1. do the side lengths of 6, 7, 5 form a right triangle? justify your answer.
2. do the side lengths of 13, 12, 5 form a right triangle? justify your answer.

Explanation:

Step1: Identify the longest side

The side lengths are \(6\), \(7\), \(5\). The longest side \(c = 7\). Let \(a = 6\) and \(b = 5\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). Calculate \(a^{2}+b^{2}\): \(6^{2}+5^{2}=36 + 25=61\). Calculate \(c^{2}\): \(7^{2}=49\).

Step3: Compare the values

Since \(61
eq49\) (i.e., \(a^{2}+b^{2}
eq c^{2}\)).

Step1: Identify the longest side

The side lengths are \(13\), \(12\), \(5\). The longest side \(c = 13\). Let \(a = 12\) and \(b = 5\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). Calculate \(a^{2}+b^{2}\): \(12^{2}+5^{2}=144 + 25 = 169\). Calculate \(c^{2}\): \(13^{2}=169\).

Step3: Compare the values

Since \(169 = 169\) (i.e., \(a^{2}+b^{2}=c^{2}\)).

Answer:

The side lengths of \(6\), \(7\), \(5\) do not form a right - triangle.