Verifying a Right Triangle In Exercises 24, (a) find the length of each side of the right triang

Verifying a Right Triangle In Exercises 24, (a) find the...

Key Concepts

Right Triangle

A right triangle is a triangle that includes one 90-degree angle. This characteristic allows for a special relationship between its sides, with two legs forming the right angle and the hypotenuse being the side opposite the right angle. Understanding right triangles is fundamental in geometry as it provides a basis for trigonometric relationships and various geometric proofs.

Pythagorean Theorem

The Pythagorean Theorem is a key principle in Euclidean geometry which states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. This theorem is widely used not only to determine side lengths but also to verify whether a given triangle is right, serving as a vital tool in both theoretical and practical applications.

Triangle Side Computation

Calculating the lengths of a triangle's sides involves applying geometric methods and algebraic formulas, such as the distance formula in coordinate geometry. Accurate computation of side lengths is essential to confirm relationships like that described by the Pythagorean Theorem, which in turn is used to verify the properties of right triangles.

Transcript

00:07 So we are given a right triangle, and in part a, we're asked to find the length of each side of this right triangle.

00:19 And in part b, we're asked to show that these lengths satisfy the pythagorean theorem.

00:31 We see that a diagram of this triangle in the cartesian plane is given in exercise 24 of this section.

00:42 I'm going to simplify the diagram by simply giving the vertices of the triangle.

00:50 So looking at our diagram, we see that our triangle has the vertices, negative 1, 1, 9 -1, and 9 -4.

01:24 We also see that this is indeed a right triangle.

01:33 Now in part a to find the length of each side of the right triangle, first of all, let's label each of these vertices.

01:45 So we'll label vertex a as negative 1 -1.

01:51 We'll label vertex 9 -1 as b and vertex 9 -4 as c.

02:04 To find the length of side a -b, we use our distance formula for the distance between two points, and therefore this is the distance from negative 1 -1 to 9 -1, which is the square root of negative 1 minus 9 squared plus 1 minus 1 squared, which is equal to the square root of negative 1 minus 9 squared is, that's negative 10, so the square root of negative 10 squared plus 1 minus 1 is 0 squared, which is still 0.

03:11 And so we have negative 10 squared is 100, and the square root of 100 is again 10.

03:18 So the length of side ab is 10.

03:26 Likewise, the length of side b, c is the distance between points b and c, or the distance between points 9 -1 and 9 -4.

03:46 And this is the square root of 9 -9 -2 -squared plus 1 -4 squared, which is the square root of 0 -2, which is 0, plus 1 -4, which is negative 3 squared.

04:07 So this is 3...

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