For The Given Triangle, The Slope Of Z X ‾ \overline{ZX} ZX Is 1 3 \frac{1}{3} 3 1 ​ , The Slope Of Z Y ‾ \overline{ZY} Z Y Is − 1 2 \frac{-1}{2} 2 − 1 ​ , And The Slope Of X Y ‾ \overline{XY} X Y Is 2. Which Statement Verifies That Triangle

Introduction

In geometry, the slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this article, we will explore the slopes of the sides of a given triangle and determine which statement verifies that the triangle is valid.

The Slopes of the Sides of a Triangle

The slope of a line is a fundamental concept in geometry. It is used to describe the steepness of a line and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the context of a triangle, the slopes of its sides are used to determine the validity of the triangle.

The Given Triangle

The given triangle has three sides: $\overline{ZX}$, $\overline{ZY}$, and $\overline{XY}$. The slope of $\overline{ZX}$ is $\frac{1}{3}$, the slope of $\overline{ZY}$ is $\frac{-1}{2}$, and the slope of $\overline{XY}$ is 2.

Understanding the Slopes of the Sides

To understand the slopes of the sides of the triangle, we need to recall the formula for calculating the slope of a line. The slope of a line is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the context of a triangle, the slopes of its sides are used to determine the validity of the triangle.

The Slope of $\overline{ZX}$

The slope of $\overline{ZX}$ is $\frac{1}{3}$. This means that for every 1 unit of horizontal change (run), there is a corresponding 3 units of vertical change (rise). This slope indicates that the line $\overline{ZX}$ is steep and has a positive slope.

The Slope of $\overline{ZY}$

The slope of $\overline{ZY}$ is $\frac{-1}{2}$. This means that for every 1 unit of horizontal change (run), there is a corresponding -2 units of vertical change (rise). This slope indicates that the line $\overline{ZY}$ is steep and has a negative slope.

The Slope of $\overline{XY}$

The slope of $\overline{XY}$ is 2. This means that for every 1 unit of horizontal change (run), there is a corresponding 2 units of vertical change (rise). This slope indicates that the line $\overline{XY}$ is steep and has a positive slope.

Determining the Validity of the Triangle

To determine the validity of the triangle, we need to check if the slopes of its sides satisfy the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

The Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In the context of the given triangle, we need to check if the sum of the lengths of any two sides is greater than the length of the third side.

Checking the Triangle Inequality Theorem

To check the triangle inequality theorem, we need to calculate the sum of the lengths of any two sides of the triangle and compare it with the length of the third side.

The Sum of the Lengths of $\overline{ZX}$ and $\overline{ZY}$

The sum of the lengths of $\overline{ZX}$ and $\overline{ZY}$ is:

$$\overline{ZX} + \overline{ZY} = \frac{1}{3} + \frac{-1}{2}$$

Simplifying the expression, we get:

$$\overline{ZX} + \overline{ZY} = \frac{2}{6} - \frac{3}{6} = \frac{-1}{6}$$

The length of $\overline{XY}$ is 2. Therefore, the sum of the lengths of $\overline{ZX}$ and $\overline{ZY}$ is less than the length of $\overline{XY}$.

The Sum of the Lengths of $\overline{ZY}$ and $\overline{XY}$

The sum of the lengths of $\overline{ZY}$ and $\overline{XY}$ is:

$$\overline{ZY} + \overline{XY} = \frac{-1}{2} + 2$$

Simplifying the expression, we get:

$$\overline{ZY} + \overline{XY} = \frac{-1}{2} + \frac{4}{2} = \frac{3}{2}$$

The length of $\overline{ZX}$ is $\frac{1}{3}$. Therefore, the sum of the lengths of $\overline{ZY}$ and $\overline{XY}$ is greater than the length of $\overline{ZX}$.

The Sum of the Lengths of $\overline{XY}$ and $\overline{ZX}$

The sum of the lengths of $\overline{XY}$ and $\overline{ZX}$ is:

$$\overline{XY} + \overline{ZX} = 2 + \frac{1}{3}$$

Simplifying the expression, we get:

$$\overline{XY} + \overline{ZX} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}$$

The length of $\overline{ZY}$ is $\frac{-1}{2}$. Therefore, the sum of the lengths of $\overline{XY}$ and $\overline{ZX}$ is greater than the length of $\overline{ZY}$.

Conclusion

In conclusion, the given triangle satisfies the triangle inequality theorem. The sum of the lengths of any two sides of the triangle is greater than the length of the third side. Therefore, the triangle is valid.

Final Answer

Introduction

In our previous article, we explored the slopes of the sides of a given triangle and determined that the triangle is valid. In this article, we will answer some frequently asked questions (FAQs) related to the slopes of the sides of a triangle.

Q&A

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do you calculate the slope of a line?

A: To calculate the slope of a line, you need to know the coordinates of two points on the line. The slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between the two points.

Q: What is the formula for calculating the slope of a line?

A: The formula for calculating the slope of a line is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Q: How do you check if a triangle is valid using the triangle inequality theorem?

A: To check if a triangle is valid using the triangle inequality theorem, you need to calculate the sum of the lengths of any two sides of the triangle and compare it with the length of the third side.

Q: What is the significance of the slopes of the sides of a triangle?

A: The slopes of the sides of a triangle are used to determine the validity of the triangle. If the sum of the lengths of any two sides of the triangle is greater than the length of the third side, then the triangle is valid.

Q: Can a triangle have a negative slope?

A: No, a triangle cannot have a negative slope. The slopes of the sides of a triangle are always positive.

Q: Can a triangle have a slope of zero?

A: No, a triangle cannot have a slope of zero. The slopes of the sides of a triangle are always positive.

Q: Can a triangle have a slope of infinity?

A: No, a triangle cannot have a slope of infinity. The slopes of the sides of a triangle are always finite.

Q: What is the relationship between the slopes of the sides of a triangle and the angles of the triangle?

A: The slopes of the sides of a triangle are related to the angles of the triangle. The slopes of the sides of a triangle are used to determine the angles of the triangle.

Q: How do you calculate the angles of a triangle using the slopes of its sides?

A: To calculate the angles of a triangle using the slopes of its sides, you need to use the formula:

$$\tan(\theta) = \frac{m_1 - m_2}{1 + m_1m_2}$$

where $\theta$ is the angle, and $m_1$ and $m_2$ are the slopes of the two sides.

Q: What is the significance of the angles of a triangle?

A: The angles of a triangle are used to determine the shape of the triangle. The angles of a triangle are used to determine the type of triangle (e.g., acute, right, obtuse).

Q: Can a triangle have a right angle?

A: Yes, a triangle can have a right angle. A right triangle has one angle that is equal to 90 degrees.

Q: Can a triangle have an obtuse angle?

A: Yes, a triangle can have an obtuse angle. An obtuse triangle has one angle that is greater than 90 degrees.

Q: Can a triangle have an acute angle?

A: Yes, a triangle can have an acute angle. An acute triangle has all angles that are less than 90 degrees.

Conclusion

In conclusion, the slopes of the sides of a triangle are used to determine the validity of the triangle. The slopes of the sides of a triangle are related to the angles of the triangle. The angles of a triangle are used to determine the shape of the triangle.