Triangle Find the measures of the angles of the triangle...

Key Concepts

Triangle

A triangle is a three-sided polygon where the sum of its interior angles is always constant. Understanding the properties of triangles, such as side lengths and angle measures, is essential for calculating unknown values within the triangle.

Coordinate Geometry

Coordinate geometry involves using a coordinate system to study and solve geometric problems. When vertices of a triangle are given in coordinate form, this approach allows for the application of algebraic methods to determine distances and angles between points.

Distance Formula

The distance formula provides a method to calculate the straight-line distance between two points in the coordinate plane. This is crucial for finding the lengths of the sides of a triangle when given their coordinates, which can then be used in further computations such as using the Law of Cosines.

Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem that applies to all triangles, not just right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles, making it an invaluable tool for finding angle measures when the side lengths are known.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to determine the measure of an angle from its cosine value. They are essential in converting the results of the Law of Cosines or vector dot product calculations into actual angle measures, typically expressed in degrees or radians.

Transcript

00:01 We want to find the measures of the angles of the triangle whose vertices are a is equal to negative 1 ,0, b is equal to 2 1, and c is equal to 1 negative 2.

00:15 So, on the board here, we're going to need to use the fact that the dot product of u and v is equal to the magnitude of u times the magnitude of v times the cosine of the angle.

00:31 And we can rearrange that to solve for theta.

00:36 And we're going to use that with each of these pairs of vectors so we can try to find the angles between them.

00:50 So if we go ahead and think about this here is the vector ab, down here the vector ac, and this here the vector b, or maybe i should say c, b.

01:14 If we were to go ahead and find each of these vectors, then we can go ahead and take the dot product of all of them with each other and then find the magnitudes of each as well and go from there.

01:31 So let's go ahead and just do that really fast.

01:34 So first to find the vector a -b, we'll remember this is going to be b -b, and so b was to 1 subtract off a, which is negative 1, 0, and then that should give us 3 -1, when we subtract each of them component was.

02:03 The vector ac is going to give us, well, we're going to have to do c minus a for this, and that's going to be 1 minus 2.

02:20 Minus minus one zero and that's going to give two negative two right and then lastly c b is going to equal to b minus c so b was 2 1 minus 2 and that there is going to give us so it's going to be 1 3 all right, so we have these now.

03:03 Now let's go ahead and find the magnitudes of each of these really quickly.

03:08 So i'm just going to put the magnitude right here.

03:11 So that's going to be equal to, well, we're going to do three squared plus one squared square root.

03:18 So that's going to be square root of 10.

03:23 The magnitude of ac, well, that's going to be two squared plus minus two squared square root.

03:32 Well that's going to be 4 plus 4 which is 8 so square root of 8 and then last like magnitude of cb well this is going to be 1 squared plus 3 squared square rooted or also the square root of so we have our magnitudes now now if we want to find any of these angles in particular we'll just need to apply the dot product as well so let's just go ahead and figure out what angles we will one first.

04:08 So i'll call angle a alpha.

04:12 So alpha is going to equal to the dot product of, well, we need the vectors a, c, and a, b.

04:26 And then we're going to divide this by the magnitudes of each of those vectors.

04:37 And we're going to take the inverse of cosine of this.

04:41 All right.

04:43 So the dot product of acab, well, we found those right here.

04:51 So that's going to be three times two, six.

04:56 Let me go ahead and write cosine first, so cosine inverse.

05:03 So that's going to be, so yeah, three times two, so six plus one times two, or so minus two.

05:17 And then all over, and then we already found these magnitudes.

05:21 So it's going to be square root of 10 times the square root of 8.

05:29 Now, let's call the other angle b.