triangle ABC has vertices A(3,4), B(4,-3) and C(-4,-1).
a. draw a sketch of the triangle
b. draw the altitude from vertex A
c. find the slope of side BC which is =-1/4
d. find the slope of the altitude from A which is 4
e. find the equation of the line that contains this altitude which y=4x-8
f. find the length of this altitude
g. find the area of the triangle.
i know from A-E BUT i do not know F and G. I have been trying my best but i did not get it can you help me pls. it is very urgent. thanks.
Of course, I'd be happy to help you with parts F and G. Let's go through each step one at a time:
f. To find the length of the altitude from vertex A, we need to calculate the distance between the point where the altitude intersects line BC and vertex A.
To start, let's find the equation of line BC using the slope and one of the given points, say point B(4, -3). We can use the point-slope form of a linear equation:
y - y1 = m(x - x1),
where m is the slope of BC (= -1/4), and (x1, y1) is a point on the line, in this case, (4, -3).
Plugging in these values, we have:
y - (-3) = (-1/4)(x - 4).
Simplifying this equation, we get:
y + 3 = (-1/4)x + 1.
Rearranging it into slope-intercept form, we have:
y = (-1/4)x - 2.
Now that we have the equation of line BC, we can find the coordinates of the point where the altitude intersects BC. This point will be the solution to the system of equations formed by the line BC and the equation of the altitude (y = 4x - 8).
Substituting the equation of the altitude into the equation of line BC, we get:
4x - 8 = (-1/4)x - 2.
Simplifying, we have:
4x + (1/4)x = 8 - 2.
Combining like terms, we get:
(17/4)x = 6.
Dividing both sides by (17/4), we can solve for x:
x = (6 * 4) / 17 = 24 / 17.
Plugging this value of x back into the equation of the altitude, we can find the corresponding y-coordinate:
y = 4 * (24 / 17) - 8 = 96 / 17 - 136 / 17 = -40 / 17.
Therefore, the coordinates of the point where the altitude intersects BC are (24/17, -40/17).
Now, we can find the length of the altitude by calculating the distance between this point and vertex A (3, 4). We can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Plugging in the values, we have:
d = sqrt((24/17 - 3)^2 + (-40/17 - 4)^2).
Simplifying this expression, we get:
d = sqrt((24/17 - 51/17)^2 + (-40/17 - 68/17)^2).
Continuing to simplify, we have:
d = sqrt((-27/17)^2 + (-108/17)^2).
d = sqrt((729/289) + (11664/289)).
d = sqrt(12393/289).
This gives us the length of the altitude.
g. Finally, to find the area of the triangle, we can use the formula:
Area = (1/2) * base * height.
Since we have the length of the altitude (height) from part f, we just need to find the length of the base. The base can be found by calculating the distance between the two given points B(4, -3) and C(-4, -1) using the distance formula:
base = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Plugging in the values, we have:
base = sqrt((-4 - 4)^2 + (-1 - (-3))^2)
base = sqrt((-8)^2 + (2)^2)
base = sqrt(64 + 4)
base = sqrt(68).
Now, we can use the formula for the area of the triangle:
Area = (1/2) * base * height.
Plugging in the values we found, we have:
Area = (1/2) * sqrt(68) * sqrt(12393/289).
Simplifying this expression gives the final answer for the area of the triangle.