Question

The converse of the Pythagorean Theorem states that if the three sides of a triangle work for the equation a2 + b2 = c2, 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 a2 + b2 = c2. 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.Part A - Since triangle 2 is a right triangle, write an equation applying the Pythagorean Theorem to the triangle.Part BSince the equations for both triangles have a2 + b2, you can think of the two equations for c2 and n2 as a system of equations. Substitute what a2 + b2 equals in the first equation for a2 + b2 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?

Answer 1

Answer and Explanation:

Given:

Part A: The equation for Triangle 2 is;

`n^2=a^2+b^2`

Part B: Since;

`\begin{gathered} a^2+b^2=c^2,\text{ and } \\ n^2=a^2+b^2 \\ Therefore, \\ c^2=n^2 \end{gathered}`

The resulting equation is c^2 = n^2

Part C: Let's take the square root of both sides from Part B;

`\begin{gathered} √(c^2)=√(n^2) \\ c=n \end{gathered}`

The resulting equation is c = n

Part D:

By taking the square root in Part B we can see that side c and side n are equal in triangle 1 and triangle 2 respectively. So the equation is true.

Part E:

The relationship, c = n, shows that triangle 1 is also a right triangle because we have that side c is equal to the hypotenuse side n of the right triangle 2.