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

Find the lengths of the sides of the triangle $P Q R .$ Is it a righttriangle? Is it an isosceles triangle?
(a) $P(3,-2,-3), Q(7,0,1), \quad R(1,2,1)$
(b) $P(2,-1,0), \quad Q(4,1,1), \quad R(4,-5,4)$.

Key Concepts

Distance Formula

The distance formula is a key tool in coordinate geometry that calculates the length between two points in space. In a three-dimensional context, the distance between two points (x?, y?, z?) and (x?, y?, z?) is given by the square root of ((x? - x?)² + (y? - y?)² + (z? - z?)²). This formula is fundamental for determining the lengths of the sides of a triangle when the vertices are given in coordinates.

Right Triangle Identification

A right triangle is characterized by one angle being exactly 90 degrees. To check for a right angle, one can use the Pythagorean theorem which states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Verifying if this condition holds for any triplet of side lengths helps in determining whether the triangle is right-angled.

Isosceles Triangle Criteria

An isosceles triangle has at least two sides of equal length. After calculating all the side lengths using the distance formula, comparing these lengths allows one to ascertain if any two are equal. This examination is essential in identifying whether the given triangle is isosceles.

Transcript

00:01 Hello and welcome.

00:03 We are looking at chapter 8, section 1, problem 7 here.

00:11 So we're asked to find the lengths of three sides, the three sides of a triangle, and then determine kind of what kind of triangle it is from that.

00:20 So the first step is to find the lengths of each of those sides using the distance formula.

00:25 So that one's going to take us a second, but it's pretty repetitive.

00:31 It's not too hard at all.

00:32 So pq, the first side of this triangle, we use our distance formula.

00:38 It's going to be the square root of, and then it's kind of the sum of the squares of the difference between the different coordinates.

00:46 So if we did 3 minus 7 squared, so that's the x.

00:53 Just put that in, just to be extra clear.

00:56 That's the difference between the two x values.

00:59 And then negative 2 minus 0 squared, and then the z's will be negative 3 minus 1.

01:14 Take the square root of that sum.

01:19 So this is the difference between the y's and the difference between z's.

01:23 So if we simplify this, 3 minus 7 is negative 4, negative 4 squared is 16.

01:33 Negative 2 minus 0 is negative 2, negative 2 squared is 4.

01:42 Negative 3 minus 1 is negative 4, negative 4 squared is 16.

01:51 So this would be 32 plus 4 is square root of 36.

01:57 So that worked out nicely for us.

01:58 That's just a distance of 6.

02:00 So the length of pq is 6.

02:07 So we've got to do that with pr and then qr.

02:11 Same process.

02:16 So this would be the square root of 3 minus 1 squared.

02:24 So this is the x values, the difference of the x values.

02:30 We'll have negative 2 minus 2 squared.

02:42 And then the z difference of the z coordinates will be negative 3 minus 1 squared.

02:54 And so this would be square root of 4 plus negative 2 minus 2 is negative 4.

03:02 So that'd be 16 when i square it...

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