Find the area of a triangle whose vertices are A(0,2), B (2,7), C (0,10).
I believe the area of the triangle would be 8
AB = sqrt((2-0)^2+(7-2)^2 = 5.39.
BC = sqrt((0-2)^2+(10-7)^2 = 3.61.
AC = sqrt(10-2)^2 = 8.
P = 5.39 + 3.61 + 8 = 17.
A^2 = (P*AB*BC*AC)/16.
a^2 = (17*5.39*3.61*8)/16 = 165.4.
A = sqrt(165.4) = 12.86.
To find the area of a triangle given its vertices, you can use the formula for the area of a triangle. The formula is:
Area = (1/2) * base * height
Here's how you can find the area of the triangle with vertices A(0,2), B(2,7), and C(0,10):
1. Identify the base and height of the triangle:
- The base of the triangle can be any one of its sides. Let's choose the base as the line segment AB with endpoints A(0,2) and B(2,7).
- The height of the triangle is the perpendicular distance from the base to the vertex C(0,10).
2. Calculate the base of the triangle:
- Use the distance formula to find the length of the base AB.
- The distance formula is: distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of A(0,2) and B(2,7) into the distance formula:
distance_AB = √[(2 - 0)^2 + (7 - 2)^2] = √(2^2 + 5^2) = √(4 + 25) = √29
3. Calculate the height of the triangle:
- To find the height, we need to find the perpendicular distance from the line segment AB to the point C(0,10).
- The height will be equal to the x-coordinate of point B or the y-coordinate of point C, as the line segment AB is perpendicular to the y-axis.
The height of the triangle is 2 units.
4. Plug the values into the area formula:
Area = (1/2) * base * height = (1/2) * √29 * 2 = √29
Therefore, the area of the triangle with vertices A(0,2), B(2,7), and C(0,10) is √29 square units.