Prove the Converse of the Pythagorean Theorem (Theorem 9.2) . Hint: Draw ABC with side lengths a

Prove the Converse of the Pythagorean Theorem (Theorem 9.2)...

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental result in geometry that applies to right triangles, stating that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem provides the basis for relating side lengths in right-angled triangles and is widely used in various mathematical contexts.

Converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem asserts that if a triangle has side lengths such that the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle must be a right triangle. This result is essential in determining the presence of a right angle based solely on side lengths.

Geometric Construction

Geometric construction involves drawing figures using specific conditions or measurements in order to explore and demonstrate mathematical properties visually. In proofs such as the Converse of the Pythagorean Theorem, constructing triangles with predetermined side lengths allows for a direct comparison between different triangle configurations and validates the necessary conditions for a right triangle.

Side Length Comparison

Comparing side lengths, particularly when constructing a triangle with a designated hypotenuse versus another triangle with a different third side, is a central method in demonstrating the equivalence conditions of the theorem. By analyzing and contrasting these lengths, one can logically deduce the nature of the triangle, ensuring that the equality required by the theorem holds only in the case of a right-angled triangle.

Transcript

00:01 So with this problem, we're asked to prove, or provide the proof for the converse of the pythagrin thero.

00:07 So in order to figure out the converse of the pythagorean theorem, or give a proof for it, we first must think about, well, what does the converse say? so if we take the normal pythagorean theorem here, oops, pythagorean theorem, at least the way that the book explains it or how we usually explain it, it says that if we have a right name, sorry, a right triangle here, a right triangle.

00:42 Then the sidelines follow a squared plus b squared is equal to c squared.

00:48 That's the normal way of how we use the pythagin and throne.

00:53 We start with a triangle that has a right angle in it, and then we get a squared equals equal to c square, which c is the length of the hypotenuse, and a and b are the length of the, or the lengths of the right triangle.

01:06 So the converse of that basically swaps the claim and the conclusion.

01:13 So the converse here, right? so this is assessing a little bit on your understanding what do you know what a converse is.

01:21 If we have a triangle that has sign length a squared plus b square equals c squared then, then we get a right triangle here.

01:31 Okay, so that's the converse.

01:33 So we have to start with the claim that a squared plus b square equals c squared here.

01:36 That's our starting point, or our claim, and our ending point here is the right triangle, which is going to be our conclusion.

01:54 Now, how do we prove the converse of the pythagrin theorem? well, let's start out with just some random triangle here.

02:03 So i just draw out a triangle that looks like this.

02:07 It looks like a right triangle, but i don't know that for sure, because i don't have any right angle.

02:14 Was explicitly drawn there.

02:16 But it does look like it.

02:18 I made it like this so that visually, it makes sense, or it's a little bit easier to understand.

02:24 But you don't have to draw it looking like a right triangle here.

02:28 Let's see how the vertices a, b, and c.

02:32 I just assign the vertices as such.

02:34 Again, completely arbitrary.

02:36 I could switch the vertices around for one and two.

02:39 Now what i'm going to do here is say, well, if i look at vertex a, then the length of the side that's opposite to that vertex a is going to be of lowercase a here.

02:50 And i could do the same thing for b and the same thing for c.

02:55 Okay.

02:57 Now, this triangle is special, right, in a sense that for triangle abc here, a squared plus b squared is equal to c squared.

03:08 It's special just like that.

03:10 That's my claim here.

03:12 So we're starting with the claim of the commerce of the pythagorean.

03:16 Theorem that a square plus b squared equals t squared okay now let me do something a little bit out of the box or weird however you want to think of it let me draw another triangle here it looks very similar it's very similar to triangle a bc except it's called triangle d e f and this triangle is special which is the right triangle here and the length of e f is a the length of fd is b and the length of ed is c.

03:56 It looks very similar to triangle abc.

03:59 Is it exactly the same? we don't know.

04:02 We don't know yet.

04:06 Actually, it would be similar triangles there.

04:09 So based on how i draw on triangle d -e -f here, triangle d -e -f is similar to triangle abc here.

04:22 Okay? now, i haven't proven anything yet.

04:26 I just made a random triangle that just so happens to have a right angle there.

04:30 And they happen to be similar to one another.

04:33 Okay? just by sss...

Please give Ace some feedback