In Exercises 59-62, the vertices of a triangle are given....

In Exercises $59-62,$ the vertices of a triangle are given. Determine whether the traingle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning.
$$
(1,2,0),(0,0,0),(-2,1,0)
$$

Key Concepts

Distance Formula

The distance formula allows you to calculate the length of a segment between any two points in Euclidean space using their coordinates. This is essential for determining the side lengths of a triangle when its vertices are given, which in turn is crucial for analyzing the triangle’s type.

Pythagorean Theorem and Its Converse

The Pythagorean Theorem relates the side lengths in a right triangle by stating that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Its converse helps determine if a triangle is right-angled, while variations in the relationships among the squared side lengths can indicate whether a triangle is acute or obtuse.

Vector Analysis and the Dot Product

Using vectors to represent the sides of a triangle allows for an analysis of the angles between sides through the dot product. A zero dot product indicates a right angle, while positive or negative dot products can be used to deduce if angles are acute or obtuse. This method offers an alternative approach to classifying triangles when vertices are provided in coordinate form.

Transcript

00:01 Welcome to numerate.

00:02 We are given a very interesting problem this time.

00:05 We are given three points.

00:07 See, a, this is 1 -20, b, which is 0 -0, and c, which is minus 2 -1 -0.

00:20 Now, we are told to find, what kind of, what are the, what kind of points are they? they will definitely form a triangle but what kind of a triangle is it a right angle triangle is it an obtuse angle is it an acute angle triangle what is it going to be so there are a number of ways to do it okay but we use intuition when we go ahead with okay so whenever you see the point this okay the origin the center of it you know whenever we multiply that with any vector we tend to get 0 why think of a into b we will get 1 into 0 plus 2 into 0 plus 0 plus 0 into 0 so this product vanishes telling that they are orthogonal to each other that means they are at right angle so if this is one point and this is one point they are orthogonal it's like exactly 90 degree between them now you see b and c same thing happens right so if this is c and this is b and this is a so they are kind of this even they are at this the last thing that will be a and c if you see for a and c also then we find that it will be 1 into minus 2 plus 2 into 1 plus 0 into zero.

02:02 That means again we are getting zero.

02:05 So therefore now our aim is to find out what kind of relation do they have.

02:13 So a triangle will always have three sides.

02:16 A b, a c, b, correct? so let's see how much is a b.

02:22 A b is nothing but root over a and b.

02:25 So 1 minus 0 whole square plus 2 minus 0.

02:31 Whole square plus 0 minus 0 full square and that would give me 2 square is 4 1 square is 1 so it's root over 5 a c a and c so that would give me positive square root of 1 minus minus 2 squared plus 2 minus 1 square plus 0 minus 0 square.

03:02 So this would give us 1 minus 1.

03:08 1 minus 2 so it will be 3.

03:12 So like 1 minus minus 2 so it's 3.

03:15 So 3 squared is 9...

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