Which statements are true? Select three options. The line $x=0$ is perpendicular to the line $y=-3$. All lines that are parallel to the $y$-axis are vertical lines. All lines that are perpendicular to the $x$-axis have a slope of 0. The equation of the line parallel to the $x$-axis that passes through the point $(2,-6)$ is $x=2$. The equation of the line perpendicular to the $y$-axis that passes through the point $(-5,1)$ is $y=1$.
Answer (2)
The line x = 0 (y-axis) is perpendicular to the line y = − 3 (horizontal line).
All lines parallel to the y -axis are vertical lines.
The equation of the line perpendicular to the y -axis that passes through the point ( − 5 , 1 ) is y = 1 .
The true statements are 1, 2, and 5. T r u e
Explanation
Analyze the statements Let's analyze each statement to determine which ones are true.
Evaluate statement 1 Statement 1: The line x = 0 is perpendicular to the line y = − 3 .
The line x = 0 represents the y-axis, which is a vertical line. The line y = − 3 is a horizontal line. Vertical and horizontal lines are perpendicular to each other. Therefore, this statement is true.
Evaluate statement 2 Statement 2: All lines that are parallel to the y -axis are vertical lines.
By definition, any line parallel to the y-axis is a vertical line. Therefore, this statement is true.
Evaluate statement 3 Statement 3: All lines that are perpendicular to the x -axis have a slope of 0.
Lines perpendicular to the x-axis are vertical lines. Vertical lines have an undefined slope, not a slope of 0. A slope of 0 corresponds to a horizontal line. Therefore, this statement is false.
Evaluate statement 4 Statement 4: The equation of the line parallel to the x -axis that passes through the point ( 2 , − 6 ) is x = 2 .
A line parallel to the x-axis is a horizontal line, which has the form y = c , where c is a constant. Since the line passes through the point ( 2 , − 6 ) , the equation of the line is y = − 6 . The given equation x = 2 represents a vertical line. Therefore, this statement is false.
Evaluate statement 5 Statement 5: The equation of the line perpendicular to the y -axis that passes through the point ( − 5 , 1 ) is y = 1 .
A line perpendicular to the y-axis is a horizontal line, which has the form y = c , where c is a constant. Since the line passes through the point ( − 5 , 1 ) , the equation of the line is y = 1 . Therefore, this statement is true.
Identify true statements The true statements are 1, 2, and 5.
Examples
Understanding the relationships between lines and their equations is crucial in various fields. For instance, in architecture, knowing that perpendicular lines form right angles is essential for designing stable and aesthetically pleasing structures. Similarly, in computer graphics, defining lines and their orientations is fundamental for rendering images and creating virtual environments. Knowing how to represent lines mathematically allows us to model and manipulate objects in a virtual space with precision.
The true statements are: 1) The line x = 0 is perpendicular to the line y = − 3 , 2) All lines that are parallel to the y -axis are vertical lines, and 5) The equation of the line perpendicular to the y -axis that passes through the point ( − 5 , 1 ) is y = 1 . Therefore, the selected options are 1, 2, and 5.
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