I have a triangle in a coordinate system(not native english) i have these three points.
A: 4,-2
B:-2,-1
C: 4,3
How do i get the height of it in relation BC?
And, how do i calculate the total area of the triangle?
Made this atm:
AB:
((-2)-4)/((-1)-2) = 2;
y-2=2(x-1)
y = 2*x;
BC:
(4+2)/(3+1) = 3/2;
y-2 = 3/2*(x-1);
y = (3/2)*x+1/2;
Took explanation out, it is in another language ;)
If you were finding slope, then you have it upside down.
slope AB = (-2 + 1)/(4+2) = -1/6
slope of BC = (3+1)/(4+2) = 4/6 = 2/3
equation of BC
y-3 = (2/3)(x-4)
3y - 9 = 2x - 8
2x - 3y + 1 = 0
distance from A(4,-2) to BC
= |2(4) - 3(-2) + 1|√(2^2 + (-3)^2)
= 15/√13
To find the height of the triangle in relation to side BC, you need to determine the equation of the line containing side BC. From your calculation, we have the equation for BC as y = (3/2)*x + 1/2.
To find the height, we need a line perpendicular to BC passing through point A(4, -2). The slope of such a line will be the negative reciprocal of the slope of BC. The slope of BC is 3/2, so the slope of the perpendicular line will be -2/3.
Using point-slope form, we can write the equation of the perpendicular line as follows:
(y - (-2)) = (-2/3) * (x - 4)
y + 2 = (-2/3)x + 8/3
y = (-2/3)x + 8/3 - 2
y = (-2/3)x + 8/3 - 6/3
y = (-2/3)x + 2/3
Now we have the equation of the line perpendicular to BC passing through point A. To find the height, we need to calculate where this line intersects the line BC.
Set the equations for the two lines equal to each other and solve for the x-coordinate:
(3/2)*x + 1/2 = (-2/3)*x + 2/3
(3/2)*x + (2/3)*x = 2/3 - 1/2
(9/6)*x + (4/6)*x = 4/6 - 3/6
(13/6)x = 1/6
x = (1/6) * (6/13)
x = 1/13
Substitute the value of x back into either equation to find the y-coordinate:
y = (-2/3) * (1/13) + 2/3
y = -(2/39) + 26/39
y = 24/39
y = 8/13
So the point of intersection of the perpendicular line and side BC is (1/13 , 8/13).
The height of the triangle in relation to side BC can be found by calculating the Euclidean distance between point A(4,-2) and the point of intersection.
Using the distance formula, the height h is given by:
h = sqrt((4 - (1/13))^2 + (-2 - (8/13))^2)
= sqrt((52/13 - 1/13)^2 + (-26/13 - 8/13)^2)
= sqrt((51/13)^2 + (-34/13)^2)
= sqrt((51^2 + 34^2)/13^2)
= sqrt((2601 + 1156)/169)
= sqrt(3757/169)
≈ sqrt(22.238)
≈ 4.712
So the height of the triangle in relation to side BC is approximately 4.712 units.
To calculate the total area of the triangle, you can use the formula for the area of a triangle which is 1/2 times the base length times the height. In this case, you can use side BC as the base and the height we just calculated. So the area is given by:
Area = (1/2) * BC * h
= (1/2) * (distance between points B and C) * 4.712
To find the distance between points B and C, you can use the distance formula again:
Distance = sqrt((4 - (-2))^2 + (3 - (-1))^2)
= sqrt((6)^2 + (4)^2)
= sqrt(36 + 16)
= sqrt(52)
≈ 7.211
So, the area of the triangle is approximately (1/2) * 7.211 * 4.712 = 16.945 square units.