Use the image to answer the question.
An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.
The graph shows the proportional relationship. Derive the equation of the line y=mx through the origin.
(1 point)
To find the slope of the line, we can use the formula:
m = (y2 - y1)/(x2 - x1)
Using the points (1,5) and (2,10), we get:
m = (10 - 5)/(2 - 1) = 5
Since the line passes through the origin, we know that the y-intercept is 0. Therefore, the equation of the line is:
y = 5x
To derive the equation of the line y=mx through the origin, we can use the slope-intercept form of a linear equation, which is y = mx + b. Since the line passes through the origin, the y-intercept (b) is 0.
Given that the line passes through the points (1, 5) and (2, 10), we can find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values from the given points, we have:
m = (10 - 5) / (2 - 1) = 5/1 = 5
So, the slope (m) of the line is 5. Since the y-intercept is 0, the equation of the line becomes:
y = 5x
To derive the equation of the line y = mx through the origin, we need to find the slope of the line.
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
In this case, the given points are (1, 5) and (2, 10). Substituting these values into the formula, we get:
m = (10 - 5) / (2 - 1)
m = 5 / 1
m = 5
Now that we know the slope (m), we can write the equation of the line in the form y = mx. Since the line passes through the origin (0, 0), the equation will be:
y = 5x
Angle ABC contains the points A(−5,3), B(−4,−2), and C(1,4). Describe the effect of the dilation of the angle with a scale factor of 8 and a center point of dilation at the origin (0,0).(1 point)
Responses
After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are A′(−40,24), B′(−32,−16), and C′(8,32).
After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are upper A prime left parenthesis negative 40 comma 24 right parenthesis , upper B prime left parenthesis negative 32 comma negative 16 right parenthesis , and upper C prime left parenthesis 8 comma 32 right parenthesis .
After dilation, the angle is eight times closer to the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle.The resulting points are A′(−58,38), B′(−12,−14), C′(18,12).
After dilation, the angle is eight times closer to the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle.The resulting points are upper A prime left parenthesis Start Fraction negative 5 over 8 End Fraction comma Start Fraction 3 over 8 End Fraction right parenthesis , upper B prime left parenthesis Start Fraction negative 1 over 2 End Fraction comma Start Fraction negative 1 over 4 End Fraction right parenthesis , upper C prime left parenthesis Start Fraction 1 over 8 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis .
After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are A′(3,11), B′(4,6), and C′(9,12).
After dilation, the angle is eight times farther from the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are upper A prime left parenthesis 3 comma 11 right parenthesis , upper B prime left parenthesis 4 comma 6 right parenthesis , and upper C prime left parenthesis 9 comma 12 right parenthesis .
After dilation, the angle is eight times closer to the point of dilation. The angle measurement remains the same. Corresponding lines (rays) are parallel between the angle and the dilated angle. The resulting points are A′(−40,24), B′(−32,−16), and C′(8,32).