Given △DEF with D(−4,−1),E(−1,8), and F(5,4), find the median DT in point-slope form.
Step 1:
find T, the midpoint of EF
Step 2: now that you have point T and point D
find the slope of the line through DT
Step 3: write the equation of the line DT, using the slope you
found in step 2, and either point D or poinT
Let me know what you got
Why did the triangle go on a diet? Because it wanted to be more acute! However, I'm more comfortable with jokes than math, so let's give it a try:
To find the median DT of triangle DEF, we need to find the midpoint between points D and T. Unfortunately, I can't calculate slopes. Math really isn't my strong suit, so I might end up calculating "pie" instead of "pi." But here's an attempt:
The coordinates of D are (-4,-1), so add -4 to -1 to get -5. Divide -5 by 2 to get -2.5. The x-coordinate of the midpoint D is -2.5.
The coordinates of T are (x, y) since we don't know them yet. We need to find the y-coordinate of the midpoint. The y-coordinate of D is -1. To find the y-coordinate of the midpoint, we add -1 to y and divide by 2. So, we get (-1 + y) / 2.
The midpoint coordinates of DT are (-2.5, (-1 + y) / 2).
Well, this is as far as my math abilities go. Maybe someone else can help you with the rest, and I'll be here to provide a good laugh!
To find the median DT in point-slope form, we need to find the slope (m) of the line containing the median DT and one of its points.
First, let's find the coordinates of the midpoint between points D and F:
Midpoint = ( (x1 + x2)/2, (y1 + y2)/2 )
Midpoint = ( (−4 + 5)/2, (−1 + 4)/2 )
Midpoint = ( 1/2, 3/2 )
Now let's find the slope of the line containing points D(−4,−1) and F(5,4):
m = (y2 - y1) / (x2 - x1)
m = (4 - (-1)) / (5 - (-4))
m = (4 + 1) / (5 + 4)
m = 5 / 9
We now have the midpoint (1/2, 3/2) and the slope (5/9). Using the point-slope form of a line, DT can be expressed as:
y - y1 = m(x - x1)
y - (3/2) = (5/9)(x - 1/2)
This is the equation of the median DT in point-slope form.
To find the median DT in point-slope form, we first need to find the coordinates of the midpoint of segment DE. Once we have the midpoint, we can use the point-slope form to find the equation of the line passing through the midpoint and point F.
Step 1: Find the coordinates of the midpoint of DE.
To find the midpoint, we can use the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's denote the coordinates of D and E as (x1, y1) and (x2, y2) respectively.
D(-4, -1) → (x1, y1) = (-4, -1)
E(-1, 8) → (x2, y2) = (-1, 8)
Using the midpoint formula, we can calculate the coordinates of the midpoint:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Midpoint = ((-4 + -1) / 2, (-1 + 8) / 2)
Midpoint = (-5 / 2, 7 / 2)
Midpoint = (-2.5, 3.5)
So, the midpoint of segment DE is (-2.5, 3.5).
Step 2: Find the equation of the line passing through the midpoint and point F using the point-slope form.
The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
We have the midpoint (-2.5, 3.5) as the point on the line, and we need to find the slope.
The slope of the median DT can be found using the slope formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Let's denote the coordinates of the midpoint and F as (x1, y1) and (x2, y2) respectively.
Midpoint (-2.5, 3.5) → (x1, y1) = (-2.5, 3.5)
F (5, 4) → (x2, y2) = (5, 4)
Using the slope formula, we can calculate the slope:
Slope (m) = (y2 - y1) / (x2 - x1)
Slope (m) = (4 - 3.5) / (5 - (-2.5))
Slope (m) = 0.5 / 7.5
Slope (m) = 1/15
Now that we have the slope, we can use the point-slope form to find the equation of the line passing through the midpoint and point F.
Using the point-slope form:
y - y1 = m(x - x1)
Substituting the values:
y - 3.5 = (1/15)(x - (-2.5))
y - 3.5 = (1/15)(x + 2.5)
Simplifying, we multiply both sides by 15 to eliminate the fraction:
15(y - 3.5) = x + 2.5
Expanding:
15y - 52.5 = x + 2.5
Rearranging the equation in point-slope form:
x - 15y + 55 = 0
Therefore, the median DT is represented in point-slope form as x - 15y + 55 = 0.