Find the lengths of the sides of the triangle P Q R . Is it a right triangle? Is it an isosceles

Find the lengths of the sides of the triangle P Q R . Is it...

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is used to determine if a triangle is right-angled by checking if the square of the longest side is equal to the sum of the squares of the other two sides. In the context of three-dimensional space, while the computation of side lengths uses the 3D distance formula, the theorem itself applies analogously to identify right angles within the triangle.

Distance Formula in Three-Dimensional Space

This concept involves calculating the distance between two points in 3D space using the formula: distance = ?[(x? - x?)² + (y? - y?)² + (z? - z?)²]. It is essential for determining the lengths of sides in triangles or any spatial configuration and forms the basis of measuring distances in geometric problems.

Triangle Classification

This concept involves identifying the type of triangle based on the lengths of its sides. For instance, a triangle can be classified as isosceles if two sides are of equal length, or as right-angled if it satisfies the Pythagorean condition, or even both. Understanding these classifications helps in analyzing the properties and behavior of triangles in geometry.

Transcript

00:02 Let's take a look at the length of sides of a couple of triangles and see if we can learn anything about the triangles once we found the lengths of their sides.

00:11 So the first triangle, triangle pqr, with vertices at p of 3, negative 2, 3, q701, and r1.

00:25 So we're going to have to use the distance formula three times.

00:28 So let's first start with the length from p to q.

00:34 We subtract x coordinates and square them.

00:37 We subtract y coordinates and square them.

00:38 We subtract z coordinates and square them.

00:41 So we're going to have 7 minus 3, subtracting x coordinates from p to q squared, plus 0 minus negative 2 squared, plus 1 minus 3 squared.

00:59 So this is going to be the square root of 4 squared, plus 2.

01:03 Squared plus negative 2 squared.

01:08 This is going to be the square root of 16 plus 4 plus 4, which is the square root of 24.

01:20 24 is 4 times 6.

01:22 So i can write this as 2 times a square root of 6, bringing the factor of 4, separating it from the factor 6, and the square root of 4 is 2.

01:33 If i look at the length from q, to r, again, same thing, subtract x coordinates, subtract y coordinates, subtract z coordinates, square each of those differences, sum them, and then take the square root.

01:48 So from q to r, the x coordinates are 1 minus 7 squared, plus y coordinates 2 minus 0 squared, plus z coordinates, 1 minus 1 squared, one squared.

02:10 I'm getting a little ahead of myself.

02:15 So i get negative six squared plus two squared plus zero squared.

02:22 It's going to give me the square root of 36 plus four plus zero.

02:32 It's going to be the square root of 40.

02:34 40 is four times 10.

02:37 Four is a perfect square so i can simplify this radical by making it two times a square root of 10.

02:45 And last but not least, the distance from p to r.

02:51 Same thing, square the x coordinate, square the difference of the x coordinates, the y coordinates, and the z coordinates.

02:57 So from p to r, i have one minus three squared plus two minus negative two squared plus one minus three squared.

03:13 This is going to give me negative two squared plus four squared.

03:19 Plus negative 2 squared.

03:21 I did not give myself enough room to write that.

03:28 So this is going to give me 4 plus 16 plus 4, which is the square root of 24, which is 24 is still 4 times 6.

03:46 The root of 4 is 2.

03:47 So this is 2 times a square root of 6.

03:52 So one of the things that we can notice right away is that this is an isoslis triangle.

03:57 We can see that because pq and pr have the same length like today.

04:20 It might be a right triangle also.

04:24 To see if it's a right triangle, we want to see if it satisfies the requirement.

04:28 The pythagorean theorem says a squared plus b squared equals c squared.

04:33 The hypotenuse is always going to be the longest side.

04:41 The longest side here was the one that had the square root of 40.

04:44 So i'm going to check is a squared, which is i'll call pqa, so 2 times a square root of 6 squared, plus the other shorter side is pr 2 times a square root of 6 squared.

05:08 And i want to see if that's equal to 2 times a square root of 10 squared...

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