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# Lengths of Triangle Sides Using the Pythagorean Theorem

## Discover, Geometrically prove, and apply the theorem

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Pythagorean Theorem

Isoceles Triangle: Is a triangle that has two sides that are equal length.

Scalene traingle: A triangle that has that all different sides

Right triangle: A triangle that is made up of a right angle.

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by \begin{align*}a^2 + b^2 = c^2\end{align*}, where \begin{align*}a\end{align*}\begin{align*}b\end{align*}, and \begin{align*}c\end{align*} are lengths of the triangle.

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Lengths of Triangle Sides Using the Pythagorean Theorem:

To find the length of one side of a triangle, given the other two sides, use the formula a2+ b2= c2

Identifying Sets of Pythagorean Triples

Pythagorean Triple: a set of three integers that make up the three sides of a right triangle for which the Pythagorean Theorem holds true

Examples of Pythagorean Triples:

3, 4, 5

5, 12, 13

7, 24, 25

11, 60, 61

Using Pythagorean Theorem to Classify Triangles:

If a2 + b2= c2 then it is a right triangle. Knowing this, what can you infer if a2 + b > c2  or if a2 + b2 < c2 ?

Using Pythagorean Theorem to Determine Distance:

We can use the Pythagorean Theorem to derive the Distance Formula.

To find the distance between two points, (x1, y1), and (x2, y2), simply plug it into the following formula.

\begin{align*}\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} = d\end{align*}

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