A line with slope 3 is 2 units away from the origin. Find the area of the triangle formed by this line and the coordinate axes.
How is this even going to be done? We are given little info but a bunch of variables. Could someone help please??
So the "legs" of the triangle would be 2 I believe. And the area of a triangle is (Length*Width)/2. So I think the area should be 2units^2. I'm not sure if that's what you were asking.
The line can pass by the origin on the right or the left, but the answer will be the same. The sloped line will be the hypotenuse of a triangle whose other two sides are segments of the x or y axes, ending at the origin.
The angles of the triangle are 71.56 degrees (arctan3), 90 degrees, and 18.44 degrees.
The length of the side opposite the small angle is 2/sin18.44 = 6.32
The length of the hypotenuse is
6.32/cos18.44 = 6.66
The area of the triangle =
(1/2)*(hypotenuse)*(height)
= 4.44 square units
I must have punched some wrong keys on my calculator
The area of the triangle is
(1/2)*(hypotenuse)*(height)
= 6.66 square units
since the height (distance of the line from the origin) is 2 units.
thank you very much it helps a lot!!!
Don't worry, I'm here to help you out with a smile! Let's solve this step by step, shall we?
We are given that the line has a slope of 3, which means for every 1 unit we move horizontally, we move 3 units vertically. Since the line is 2 units away from the origin, we can form a right-angled triangle.
To find the area of the triangle, we need to find the base and height of the triangle. The base will be the distance from the y-axis to the point on the x-axis where the line intersects. The height will be the distance from the x-axis to the point on the y-axis where the line intersects.
Since the line has a slope of 3, when it intersects with the x-axis, the y-coordinate will be 0. Using the slope-intercept form of a line, we can set up the equation y = mx + b, where m is the slope and b is the y-intercept. Since the line passes through the origin, the equation becomes y = 3x.
Setting y to 0, we find that x = 0. So the point of intersection on the x-axis is (0, 0).
Similarly, when the line intersects with the y-axis, the x-coordinate will be 0. Plugging in x = 0 into the equation y = 3x, we find that y = 0. So the point of intersection on the y-axis is (0, 0).
Thus, the base and height of the triangle are both 2 units.
The formula for the area of a triangle is 1/2 * base * height. Plugging in the values, we get 1/2 * 2 * 2 = 2 square units.
So, the area of the triangle formed by the line and the coordinate axes is 2 square units. Voila!
I hope that brought a smile to your face while learning. If you have any more math questions or need help with anything else, feel free to ask!
To find the area of the triangle formed by the line with slope 3 (let's call it Line L) and the coordinate axes, we can use the following steps:
1. Find the points where Line L intersects the x-axis and y-axis.
2. Calculate the lengths of the two sides of the triangle formed by Line L and the x-axis.
3. Calculate the length of the third side of the triangle formed by Line L and the y-axis.
4. Use the formula for the area of a triangle to calculate the area.
Step 1: Finding the points of intersection
We know that Line L has a slope of 3. Since the line is 2 units away from the origin, we can use its slope to find the values of x and y coordinates of the points where it intersects the axes.
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Since Line L passes through the origin (0, 0), its equation is y = 3x.
To find the point where Line L intersects the x-axis, substitute y = 0 into the equation: 0 = 3x. Solving for x, we get x = 0. This means that Line L intersects the x-axis at the origin (0, 0).
To find the point where Line L intersects the y-axis, substitute x = 0 into the equation: y = 3(0). Solving for y, we get y = 0. This means that Line L intersects the y-axis at the origin (0, 0).
Step 2: Calculating the length of the base
The length of the base of the triangle formed by Line L and the x-axis is the distance between the two points: (0, 0) (the origin) and the other point of intersection.
Since we know that the other point of intersection has coordinates (x, 0), we can directly calculate the length of the base by using the distance formula:
distance = |x1 - x2|
In this case, x1 = 0 and x2 = x (the x-coordinate of the point of intersection). Therefore, the length of the base is simply the absolute value of x.
Step 3: Calculating the length of the height
The length of the height of the triangle formed by Line L and the y-axis is the distance between the two points: (0, 0) (the origin) and the other point of intersection.
Since we know that the other point of intersection has coordinates (0, y), we can directly calculate the length of the height by using the distance formula:
distance = |y1 - y2|
In this case, y1 = 0 and y2 = y (the y-coordinate of the point of intersection). Therefore, the length of the height is simply the absolute value of y.
Step 4: Calculating the area of the triangle
Now that we have the length of the base and the height, we can use the formula for the area of a triangle:
area = (base * height) / 2
Substitute the values we have calculated for the base and height, and compute the area using this formula.
I hope this explanation helps you understand how to approach and solve this problem.