Lydia graphed $\Delta XYZ$ at the coordinates $X(0,-4), Y(2,-3)$, and $Z(2,-6)$. She thinks $\Delta XYZ$ is a right triangle. Is Lydia's assertion correct?
A. Yes; the slopes of $\overline{ XY }$ and $\overline{ XZ }$ are the same.
B. Yes; the slopes of $\overline{XY}$ and $\overline{XZ}$ are opposite reciprocals.
C. No; the slopes of $\overline{XY}$ and $\overline{XZ}$ are not the same.
D. No; the slopes of $\overline{ XY }$ and $\overline{ XZ }$ are not opposite reciprocals.
Answer (1)
Calculate the slope of each side of the triangle using the coordinates of the vertices.
Check if any two slopes are negative reciprocals of each other, which would indicate perpendicularity.
Determine that no two sides are perpendicular.
Conclude that the triangle is not a right triangle and Lydia's assertion is incorrect. No; the slopes of X Y and XZ are not opposite reciprocals.
Explanation
Problem Analysis The coordinates of the vertices of triangle XYZ are X(0, -4), Y(2, -3), and Z(2, -6). We need to determine if triangle XYZ is a right triangle by checking if any two sides are perpendicular. We will do this by calculating the slopes of each side and checking if any two slopes are negative reciprocals of each other.
Slope Formula The slope of a line segment between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 Let's calculate the slopes of the sides of the triangle.
Calculate Slopes The slope of side XY is: m X Y = 2 − 0 − 3 − ( − 4 ) = 2 1 The slope of side XZ is: m XZ = 2 − 0 − 6 − ( − 4 ) = 2 − 2 = − 1 The slope of side YZ is: m Y Z = 2 − 2 − 6 − ( − 3 ) = 0 − 3 Since the denominator is zero, the slope of YZ is undefined. This means that YZ is a vertical line.
Check for Perpendicularity Now we need to check if any two slopes are negative reciprocals of each other. Two lines are perpendicular if the product of their slopes is -1.
Let's check XY and XZ: m X Y ⋅ m XZ = 2 1 ⋅ ( − 1 ) = − 2 1 = − 1 So, XY and XZ are not perpendicular.
Since YZ is a vertical line, we need to check if either XY or XZ is a horizontal line. A horizontal line has a slope of 0. Since neither m X Y nor m XZ is 0, neither XY nor XZ is horizontal. However, a line perpendicular to a vertical line has a slope of 0. A line perpendicular to a line with slope 0 is undefined. Since the slope of YZ is undefined, we need to check if YZ is perpendicular to XY or XZ. For YZ to be perpendicular to another line, that line must be horizontal (slope of 0). Since neither XY nor XZ has a slope of 0, YZ is not perpendicular to either of them. However, we can check if XZ is perpendicular to YZ. Since YZ is vertical, a perpendicular line must be horizontal, meaning its slope is 0. Since the slope of XZ is -1, XZ is not perpendicular to YZ.
However, since YZ is a vertical line at x=2, and XZ has a slope of -1, it is not perpendicular to YZ. The line perpendicular to YZ must be a horizontal line. Since XZ is not horizontal, it is not perpendicular to YZ. Since XY has a slope of 1/2, it is not horizontal, so it is not perpendicular to YZ. Therefore, no two sides are perpendicular.
Conclusion Since none of the pairs of sides are perpendicular, triangle XYZ is not a right triangle. Therefore, Lydia's assertion is incorrect. The slopes of X Y and XZ are not opposite reciprocals.
Examples
In architecture, determining if walls meet at right angles is crucial for structural integrity. By calculating the slopes of the walls and checking if they are negative reciprocals, architects can ensure that the building is stable and meets safety standards. This principle extends to various fields, including carpentry, engineering, and even creating accurate maps.
