The x-axis, the y-axis, and the line through the point (1, 3) having slope -3 form a triangle, find the area.
Is the answer 6
And how do u solve it
slope = -3 means
y = -3x + b
If (1,3) is on the line.
3 = -3(1)+b
b = 6
So, the line is y = -3x+6
This line has x- and y-intercepts at (2,0) and (0,6)
So, the triangle has base 2 and height 6, giving area = 6
You are correct.
To find the area of the triangle formed by the x-axis, the y-axis, and the line through the point (1, 3) with a slope of -3, we can use the formula for the area of a triangle.
The formula for the area of a triangle is given by:
Area = 1/2 * base * height
In this case, the base of the triangle is the length of the x-axis, which is the distance between the points (0, 0) and (1, 0). We can calculate the distance using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we have:
Distance = sqrt((1 - 0)^2 + (0 - 0)^2)
= sqrt(1^2 + 0^2)
= sqrt(1)
= 1
The height of the triangle is the distance between the point (1, 3) and the line formed by the x-axis. To find the height, we need to find the equation of the line.
The equation of a line with slope m and passing through the point (x1, y1) is given by:
y - y1 = m(x - x1)
In this case, the point (1, 3) lies on the line, and the slope is -3. Plugging in the values, we have:
y - 3 = -3(x - 1)
Simplifying the equation, we get:
y - 3 = -3x + 3
Rearranging the equation, we have:
y = -3x + 6
Now we can find the height by calculating the y-coordinate of the point where the line intersects the y-axis. Plugging in x=0, we have:
y = -3(0) + 6
= 6
So the height of the triangle is 6.
Now we can plug in the values of the base and height into the formula for the area:
Area = 1/2 * base * height
= 1/2 * 1 * 6
= 3
Therefore, the area of the triangle formed by the x-axis, the y-axis, and the line through the point (1, 3) with a slope of -3 is 3.