Find the intersection of those two lines r=(2,9)+s(1,1) and r= (-3,2) + t(1,2) and then find the area of the triangle whose vertices are the intersection point, s = 0 on L1 and t = 3 on L2.
I already found the point of intersection. I just need to know how to go about solving for the area of the triangle whose vertices are the intersection point, s=0 on L1 and t=3 on L2.
When s = 0 , we get the point (2,9)
when t=3, we get the point (0,8)
Using your intersection point that you already found, find the lengths of the 3 line segments using the "distance between two points" formula.
To find the area, use Heron's Formula, which says
Area = √(s(s-a)(s-b)(s-c)) where s = 1/2 the perimeter, and a, b, and c are the lengths of the three sides.
Heron's Formula:
http://www.mathwarehouse.com/geometry/triangles/area/herons-formula-triangle-area.php
Happy calculating !
By "find the length of the 3 line segments" you're referring to the points on each side of the triangle which is (2,9), (0,8) and (-1,6) (intersecting point)?
yes, those three point define three line segments.
To find the area of the triangle with the given vertices, you can use the formula for the area of a triangle in vector form.
First, we need to find two vectors that represent the sides of the triangle.
1. Side 1: The vector between the intersection point and the point when s = 0 on L1. We have the intersection point, so now we need to find the corresponding point on L1 when s = 0.
Substitute s = 0 into the equation of L1:
r = (2,9) + 0(1,1)
r = (2,9)
The vector between the intersection point and the point when s = 0 on L1 is:
v1 = (2,9) - intersection point
2. Side 2: The vector between the intersection point and the point when t = 3 on L2. We already have the intersection point, so we need to find the corresponding point on L2 when t = 3.
Substitute t = 3 into the equation for L2:
r = (-3,2) + 3(1,2)
r = (0,8)
The vector between the intersection point and the point when t = 3 on L2 is:
v2 = (0,8) - intersection point
Now that we have the two sides of the triangle, we can calculate the area using the cross product of the two vectors.
Area = 0.5 * |v1 x v2|
To calculate the cross product:
1. Compute the cross product of the two vectors:
v1 x v2 = (v1y * v2z - v2y * v1z, v2x * v1z - v1x * v2z, v1x * v2y - v2x * v1y)
2. Calculate the magnitude of the cross product:
|v1 x v2| = sqrt((v1y * v2z - v2y * v1z)^2 + (v2x * v1z - v1x * v2z)^2 + (v1x * v2y - v2x * v1y)^2)
Finally, use the formula for the area:
Area = 0.5 * |v1 x v2|
Plug in the values of the vectors v1 and v2, and calculate the area to find the answer.