Lesson Objectives

- Demonstrate an understanding of the Pythagorean Formula
- Learn how to determine if three points are the vertices of a right triangle

## How to Determine if Three Points are the Vertices of a Right Triangle

In the last lesson, we learned how to find the distance between two points using the distance formula. The distance formula is a direct application of the Pythagorean Formula: $$a^2 + b^2=c^2$$ Where a and b are the two shorter sides, known as legs, and c is the longest side, known as the hypotenuse: We can use the distance formula and the Pythagorean Formula to determine if three points are the vertices of a right triangle.

Example #1: Determine if the given points are the vertices of a right triangle. $$(-7,4), (5,9), (5,4)$$ There are three distances we need to find. These will make up the a, b, and c from our formula. We will then plug into the formula and determine if we have a right triangle. We will use d

Example #1: Determine if the given points are the vertices of a right triangle. $$(-7,4), (5,9), (5,4)$$ There are three distances we need to find. These will make up the a, b, and c from our formula. We will then plug into the formula and determine if we have a right triangle. We will use d

_{1}to notate the distance between (5,9) and (5,4). Recall our distance formula: $$d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ $$d_1=\sqrt{(5 - 5)^2 + (4 - 9)^2}$$ $$d_1=\sqrt{(-5)^2}=\sqrt{25}=5$$ We notice that the x-coordinate is the same in each case, this means our distance formula will just come down to the absolute value of the difference between the y-coordinates. $$d_1=|4 - 9|=|-5|=5$$ Now, let's look at the distance between two other points: We will use d_{2}to notate the distance between (-7,4) and (5,4). Here the y-coordinates are the same. Again, we can use the same shortcut as above. $$d_2=|-7 - 5|=|-12|=12$$ Let's look at the distance between the last pair of points: We will use d_{3}to notate the distance between (-7,4) and (5,9). $$d_3=\sqrt{(5 - (-7))^2 + (9 - 4)^2}$$ $$d_3=\sqrt{169}=13$$ We think about the two smaller distances as a and b, and the largest distance as c. We plug into the formula and see if the equality is true: a = 5, b = 12, c = 13 $$a^2 + b^2=c^2$$ $$5^2 + 12^2=13^2$$ $$25 + 144=169$$ $$169=169$$ Since this statement is true, we know these three points are the vertices of a right triangle. Example #2: Determine if the given points are the vertices of a right triangle. $$(-5, 3), (-4, 8), (-7, 5)$$ There are three distances we need to find. These will make up the a, b, and c from our formula. We will then plug into the formula and determine if we have a right triangle. We will use d_{1}to notate the distance between (-7,5) and (-5,3). Recall our distance formula: $$d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ $$d_1=\sqrt{(-5 - (-7))^2 + (3 - 5)^2}$$ $$d_1=\sqrt{8}=2\sqrt{2}$$ Now, let's look at the distance between two other points: We will use d_{2}to notate the distance between (-7,5) and (-4,8). $$d_2=\sqrt{(-4 - (-7))^2 + (8 - 5)^2}$$ $$d_2=\sqrt{18}=3\sqrt{2}$$ Let's look at the distance between the last pair of points: We will use d_{3}to notate the distance between (-4,8) and (-5,3). $$d_3=\sqrt{(-5 - (-4))^2 + (3 - 8)^2}$$ $$d_3=\sqrt{26}$$ We think about the two smaller distances as a and b, and the largest distance as c. We plug into the formula and see if the equality is true: $$a=2\sqrt{2}$$ $$b=3\sqrt{2}$$ $$c=\sqrt{26}$$ $$a^2 + b^2=c^2$$ $$(2\sqrt{2})^2 + (3\sqrt{2})^2=(\sqrt{26})^2$$ $$8 + 18=26$$ $$26=26$$ Since this statement is true, we know these three points are the vertices of a right triangle.#### Skills Check:

Example #1

Determine if the points are the vertices of a right triangle.

Please choose the best answer. $$(3,9), (6,9), (3,13)$$

A

Yes

B

No

Example #2

Determine if the points are the vertices of a right triangle.

Please choose the best answer. $$(-5, 4), (7, 4), (2, 17)$$

A

Yes

B

No

Example #3

Determine if the points are the vertices of a right triangle.

Please choose the best answer. $$(2,9), (5,12), (8,15)$$

A

Yes

B

No

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