Verifying a Polygon, show that the points form the vertices of the indicated polygon. Right tria

Verifying a Polygon, show that the points form the vertices...

Key Concepts

Pythagorean Theorem

The Pythagorean theorem 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. This theorem is a fundamental tool for confirming the right-angle property of a triangle by comparing side lengths.

Slope Calculation

Calculating the slope of the line segments connecting any two points is a key technique in coordinate geometry. In the context of verifying right triangles, determining the slopes allows one to check perpendicularity, since two lines are perpendicular if the product of their slopes is -1.

Right Triangle Properties

A right triangle is a type of polygon with three sides where one of the angles is exactly 90 degrees. The properties of a right triangle are essential in many areas of mathematics and include the characteristics that one side is perpendicular to another, and the relationships between the lengths of the sides are dictated by the Pythagorean theorem.

Polygon Verification

This concept involves confirming that a set of points can serve as vertices forming a closed shape. In coordinate geometry, verifying a polygon means ensuring that the points connect in a specific order without any overlap or gaps and that they form the intended geometric figure.

Distance Formula

The distance formula is used in coordinate geometry to calculate the length between two points. It is derived from the Pythagorean theorem and is fundamental when verifying whether the sides of a triangle have lengths that satisfy certain geometric conditions, such as those needed for a right triangle.

Transcript

00:02 Hello, in this problem i'll be guiding you through the steps to prove that these particular vertices create a right triangle using distance.

00:11 Please take a moment to plot out the graph if you haven't already so that you have it for your own reference.

00:17 In this problem, we will be using the distance formula to confirm that pythagorean's theorem a squared plus b squared equals c squared stands true.

00:27 We're going to start off by using the distance formula, which is d equals the square root of x sub 2 minus x sub 1 squared, plus y sub 2 minus y sub 1 squared.

00:40 Depending on which way the vector is going will determine which x's and y's are sub 1s or sub 2s.

00:46 I'm going to start off by figuring out vector ab.

00:50 You're going to want to take the distance formula and plug in the corresponding variables.

00:55 Since the vector starts with a and ends with b, then a will have both x sub 1 and y sub 1 and b will have x sub 2 and y sub 2.

01:03 Had it been vector b -a, then the sub -1s and sub -2s would simply swap places in the equation.

01:11 Going forward, plugging in our figures, we'll come up with vector ab, the square root of 4 minus 2 squared, plus 0 minus 1 squared, which would alternatively be filled in as the square root of 4 plus 1 after you do the math, which equals the square root of 5.

01:46 And we can leave it like that, and you don't have to worry, because later it'll be easy to solve it once we put it into the pythagorean's theorem.

01:55 Next, we will have to figure out what vectors bc and c .r.

02:00 Again, please take note of the direction of the vector, as it will be important in determining which coordinates have sub 1s or sub 2s.

02:07 Remember that negative numbers will always yield a positive result because the negatives will cancel out each other.

02:14 So continuing, we will move to vector bc and plug in our corresponding values in the distance formula again.

02:22 Which would give you negative 1 minus 4 squared plus negative 5 minus 0 squared, which would alternatively give you 25 plus 25...

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