Here are some more questions I'm having trouble with. Please walk me through them so I can fully understand them.
#2: Let A be (5,9), B be (-3,-5), and C be (1,1). The median of a triangle connects a vertex of a triangle to the midpoint of the opposite side. *Another note: would you be able to explain the median of a triangle more fully? I don't get what my teacher meant by the midpoint on the other side* For example, the median of triangle ABC from vertex A connects A to the midpoint of line segment BC (I still don't understand it).
a. Find an equation describing the line that contains the median from A to the midpoint of line segment BC.
b. Find an equation describing the line that contains the median from B to the midpointof line segment AC.
c. What point (x,y) is on both of the lines you found in parts a and b (I might be able to figure this out if you explain a and b well).
d. Find an equation describing the line that contains the median A to the midpoint of line segment AB.
Please explain this very thoroughly so I can understand it. I am only in 7th grade algebra so there are a lot of terms I still don't understand.
Thank you so much!
Of course! I'll be happy to explain the concept of medians in a triangle and how to solve the given questions step by step.
The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. It is called a median because it divides the opposite side into two equal segments.
Now let's solve the questions step by step:
a. Find an equation describing the line that contains the median from A to the midpoint of line segment BC.
To find the midpoint of line segment BC, we need to use the coordinates of B and C. The midpoint formula states that the coordinates of the midpoint (x, y) can be found by averaging the x-coordinates of B and C, and then averaging the y-coordinates of B and C.
The x-coordinate of B is -3 and the x-coordinate of C is 1.
(x₁ + x₂)/2 = (-3 + 1)/2 = -2/2 = -1.
The y-coordinate of B is -5 and the y-coordinate of C is 1.
(y₁ + y₂)/2 = (-5 + 1)/2 = -4/2 = -2.
So, the midpoint of line segment BC is (-1, -2).
Now we can find the equation of the line that passes through A(5, 9) and the midpoint of BC (-1, -2) using the point-slope form of a linear equation.
The formula for the point-slope form is: y - y₁ = m(x - x₁), where m is the slope of the line.
First, let's find the slope (m) of the line:
m = (y₂ - y₁)/(x₂ - x₁) = (-2 - 9)/(-1 - 5) = -11/-6 = 11/6.
Using the point-slope form, we substitute the coordinates of point A (5, 9):
y - 9 = (11/6)(x - 5).
And that's the equation describing the line that contains the median from A to the midpoint of line segment BC.
b. Find an equation describing the line that contains the median from B to the midpoint of line segment AC.
Similarly, we need to find the midpoint of line segment AC. The x-coordinate of A is 5, and the x-coordinate of C is 1.
(x₁ + x₂)/2 = (5 + 1)/2 = 6/2 = 3.
The y-coordinate of A is 9, and the y-coordinate of C is 1.
(y₁ + y₂)/2 = (9 + 1)/2 = 10/2 = 5.
Therefore, the midpoint of line segment AC is (3, 5).
Now, we can find the equation of the line that passes through B(-3, -5) and the midpoint of AC (3, 5).
Using the slope formula:
m = (y₂ - y₁)/(x₂ - x₁) = (5 - (-5))/(3 - (-3)) = 10/6 = 5/3.
Using the point-slope form and substituting the coordinates of point B (-3, -5):
y - (-5) = (5/3)(x - (-3)).
That's the equation describing the line that contains the median from B to the midpoint of line segment AC.
c. To find the point that lies on both lines described in parts a and b, we need to solve the two equations simultaneously.
Setting equation (a) and equation (b) equal to each other, we have:
(11/6)(x - 5) + 9 = (5/3)(x + 3) - 5.
We simplify the equation:
11(x - 5)/6 + 9 = 5(x + 3)/3 - 5.
Multiplying both sides by 6 to eliminate the fractions, we get:
11(x - 5) + 54 = 10(x + 3) - 30.
Expanding and simplifying both sides of the equation:
11x - 55 + 54 = 10x + 30 - 30.
Simplifying further:
11x - 1 = 10x.
Now we solve for x:
11x - 10x = 1,
x = 1.
To find the y-coordinate, we substitute the value of x into either of the equations from parts a or b.
Using equation (a):
y - 9 = (11/6)(1 - 5).
Simplifying:
y - 9 = (11/6)(-4),
y - 9 = -22/6,
y - 9 = -11/3.
Now, solve for y:
y = -11/3 + 9,
y = -11/3 + 27/3,
y = 16/3.
Therefore, the point (x, y) that lies on both lines is (1, 16/3).
d. Find an equation describing the line that contains the median from A to the midpoint of line segment AB.
Since A is the vertex itself, the midpoint of line segment AB will simply be the coordinates of B(-3, -5).
Using the slope formula:
m = (y₂ - y₁)/(x₂ - x₁) = (-5 - 9)/(-3 - 5) = -14/-8 = 7/4.
Using the point-slope form and substituting the coordinates of point A (5, 9):
y - 9 = (7/4)(x - 5).
That's the equation describing the line that contains the median from A to the midpoint of line segment AB.
I hope this thorough explanation helps you understand how to find the equations of medians in a triangle! Let me know if you have any further questions.