Finding Slopes And Classifying Triangles: A Step-by-Step Guide

Oct 17, 2025 by ADMIN 63 views

Hey math enthusiasts! Today, we're diving into the fascinating world of coordinate geometry. We'll be using the coordinates of a triangle's vertices to calculate slopes and, ultimately, classify the triangle. This is super important because understanding slopes helps us determine the relationships between lines and angles, which is a fundamental concept in geometry. So, grab your pencils, and let's get started! We'll be working with a specific triangle, $\triangle XYZ$ , and breaking down the process step-by-step. Let's make this fun and easy to understand, alright?

Calculating Slopes: The Heart of the Matter

Alright, guys, first things first: we need to understand what a slope is. In simple terms, the slope of a line measures its steepness or the rate at which it rises or falls. Think of it like this: if you're hiking up a hill, the steeper the hill, the greater the slope. Mathematically, the slope ( $m$ ) of a line passing through two points, $(x_1, y_1)$ and $(x_2, y_2)$ , is calculated using the following formula: $m = (y_2 - y_1) / (x_2 - x_1)$ . Remember this formula! It's the key to unlocking our triangle's secrets.

Now, let's get down to business and calculate the slopes of the sides of our triangle, $\triangle XYZ$ . The coordinates of the vertices are given as: $X(-5, 5)$ , $Y(-3, -2)$ , and $Z(4, 0)$ . We'll find the slope of each side: $\overline{XZ}$ , $\overline{YZ}$ , and $\overline{XY}$ .

Slope of $\overline{XZ}$

To find the slope of $\overline{XZ}$ , we'll use the coordinates of points $X$ and $Z$ . Point $X$ is $(-5, 5)$ and point $Z$ is $(4, 0)$ . Let's plug these values into our slope formula: $m_{XZ} = (0 - 5) / (4 - (-5))$ . Simplifying this, we get $m_{XZ} = -5 / 9$ . So, the slope of $\overline{XZ}$ is $-5/9$ . This tells us that as we move from left to right along the line segment $\overline{XZ}$ , it slopes downwards. Remember the importance of correctly substituting the values in the slope formula; a simple error can lead to a completely different result. Understanding how to calculate slopes is crucial, as it forms the basis for many other geometric calculations.

Slope of $\overline{YZ}$

Next up, we need to find the slope of $\overline{YZ}$ . We'll use the coordinates of points $Y$ and $Z$ . Point $Y$ is $(-3, -2)$ and point $Z$ is $(4, 0)$ . Plugging these into the slope formula, we get: $m_{YZ} = (0 - (-2)) / (4 - (-3))$ . Simplifying, we have $m_{YZ} = 2 / 7$ . Therefore, the slope of $\overline{YZ}$ is $2/7$ . This positive slope indicates that as we move from left to right along the line segment $\overline{YZ}$ , it slopes upwards. See how changing the points changes the slope? It's like a different path, each having a unique characteristic.

Slope of $\overline{XY}$

Finally, let's calculate the slope of $\overline{XY}$ . We'll use the coordinates of points $X$ and $Y$ . Point $X$ is $(-5, 5)$ and point $Y$ is $(-3, -2)$ . Using the slope formula, we get: $m_{XY} = (-2 - 5) / (-3 - (-5))$ . Simplifying this, we have $m_{XY} = -7 / 2$ . Thus, the slope of $\overline{XY}$ is $-7/2$ . This large negative slope tells us that the line segment $\overline{XY}$ slopes downwards quite steeply from left to right. Now that we've found the slopes of all three sides, we are equipped to classify the triangle! Now that we have calculated all the slopes, we are well on our way to understanding the properties of the triangle. Understanding slopes is a fundamental skill in geometry and will aid in further problem-solving.

Classifying the Triangle: Putting it All Together

Alright, folks, we've done the hard work, and now it's time to classify our triangle. The classification of a triangle depends on its sides and angles. Based on the sides, triangles can be classified as scalene (no equal sides), isosceles (two equal sides), or equilateral (all sides equal). Based on the angles, triangles can be classified as acute (all angles less than 90 degrees), obtuse (one angle greater than 90 degrees), or right (one angle equal to 90 degrees). To classify our $\triangle XYZ$ , we can use the slopes we've just calculated.

Analyzing the Slopes

Let's revisit the slopes we found earlier: $m_{XZ} = -5/9$ , $m_{YZ} = 2/7$ , and $m_{XY} = -7/2$ . Now, the key here is to determine if any of the sides are perpendicular. Remember that two lines are perpendicular if the product of their slopes is -1. This is a crucial concept. Let's see if we have any perpendicular sides. If any two sides are perpendicular, the triangle is a right triangle. If no sides are perpendicular, we will need to calculate the side lengths and angles to make a proper classification. We can tell that none of the slopes are negative reciprocals of each other by a simple observation, meaning no product will equal -1. Therefore, $\triangle XYZ$ is not a right triangle. Since we have to determine the sides and angles, we can go ahead and determine those for more accuracy.

Checking for Right Angles and Side Lengths

For a right triangle, we'd look for slopes that are negative reciprocals of each other (like 2 and -1/2). Since we don't have any, we know our triangle isn't a right triangle. To classify the triangle more precisely, we'll calculate the lengths of its sides and maybe even measure the angles. Let's calculate the side lengths using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Length of XY: $d_{XY} = \sqrt{(-3 - (-5))^2 + (-2 - 5)^2} = \sqrt{2^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}$ .
Length of YZ: $d_{YZ} = \sqrt{(4 - (-3))^2 + (0 - (-2))^2} = \sqrt{7^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}$ .
Length of XZ: $d_{XZ} = \sqrt{(4 - (-5))^2 + (0 - 5)^2} = \sqrt{9^2 + (-5)^2} = \sqrt{81 + 25} = \sqrt{106}$ .

Final Classification

Since sides XY and YZ are equal in length (both $\sqrt{53}$ ), we know that $\triangle XYZ$ is an isosceles triangle. We can further find the angles using the slopes and the arctangent function, but for our purposes, classifying it based on sides is sufficient. We have successfully determined that $\triangle XYZ$ is an isosceles triangle because two of its sides are equal. We have learned how to use slopes to classify triangles and to relate the equations to the coordinates. We started with the slopes and then determined side lengths. Bravo, guys, we made it! Understanding and calculating slopes is a fundamental skill in geometry. Keep practicing these types of problems, and you'll become a geometry whiz in no time!

Conclusion: Mastering Slopes and Triangles

So, there you have it, friends! We've successfully calculated the slopes of the sides of $\triangle XYZ$ and classified it as an isosceles triangle. This process is a fantastic example of how coordinate geometry allows us to analyze and understand geometric figures using algebraic methods. Remember, the slope formula is your best friend! Keep practicing, and you'll become a pro at finding slopes and classifying triangles. Understanding slopes is a key concept that underpins much of advanced math and science. From here, you can move on to other geometric challenges, such as finding the area of the triangle, determining its angles, and working with more complex shapes. Great job everyone! You have successfully learned how to use coordinate geometry to solve problems and understand the properties of triangles. Keep practicing, and you'll find that these concepts become second nature. You've got this!