Question

Explaining the Converse of the Pythagorean TheoremThe converse of the Pythagorean Theorem states that if the three sides of a triangle work for the equation a^2 + b^2 = c^2, then the triangle is a right triangle. To prove this, you can use what’s called a proof by contradiction. That is, you can prove something is true because it cannot be false.Start by assuming a triangle is not a right triangle and the sides work for the equation a^2 + b^2 = c^2. Here is a diagram of the triangle. Keep this diagram window open as you work on the tasks in this section.Now, create a right triangle with legs a and b. Call the hypotenuse n. Here is a diagram of the triangle. Keep this diagram window open as you work on the tasks in this section.questionsPart ASince triangle 2 is a right triangle, write an equation applying the Pythagorean Theorem to the triangle.Part BSince the equations for both triangles have a^2 + b^2, you can think of the two equations for c^2 and n^2 as a system of equations. Substitute what a^2 + b^2 equals in the first equation for a^2 + b^2 in the second equation. After you substitute, what equation do you get?Part CNow, take the square root of both sides of the equation from part B and write the resulting equation.Part DIs there any way for this equation to be true? How?Part EWhat does this show about the relationship between the two triangles?Part FDoes this mean that triangle 1 is a right triangle? Why or why not?

Answer 1

Part A: Since triangle 2 is a right triangle, write an equation applying the Pythagorean Theorem to the triangle.

Triangle 2 has the following sides: a, b and n

Writing it into an equation will be:

`\text{ a}^2\text{ + b}^2\text{ = n}^2`

The answer is a² + b² = n²

Part B: Since the equations for both triangles have a^2 + b^2, you can think of the two equations for c^2 and n^2 as a system of equations. Substitute what a^2 + b^2 equals in the first equation for a^2 + b^2 in the second equation. After you substitute, what equation do you get?

Equation 1 (Triangle 1): a² + b² = c²

Equation 2 (Triangle 2): a² + b² = n²

Substitute what a^2 + b^2 equals in the first equation for a^2 + b^2 in the second equation, it will be:

`\text{ a}^2\text{ + b}^2\text{ = n}^2`
`\text{ c}^2\text{ = n}^2`

The answer is c² = n²

Part C : Now, take the square root of both sides of the equation from part B and write the resulting equation.

`\text{ c}^2\text{ = n}^2`
`\text{ }√(c^2)\text{ = }√(n^2)`
`\text{ c = n}`

The answer is c = n