Given Δabc with a(–3, 4), b(5, 8), and c(2, –2), write the equation of the line containing midsegment in standard form, where x is the midpoint of and z is the midpoint of .

Question

1) 6x - 5y = -24
2) 2x + 5y = 32
3) 2x - 5y = -28
4) 6x + 5y = 36

Answer 1

To find the midpoint of segment AB, we can use the midpoint formula:

Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Using the coordinates of A(-3, 4) and B(5, 8), we get:

Midpoint of AB = ( (-3 + 5)/2 , (4 + 8)/2 )
= ( 2/2 , 12/2 )
= ( 1 , 6 )

Similarly, to find the midpoint of segment BC, we can use the midpoint formula:

Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Using the coordinates of B(5, 8) and C(2, -2), we get:

Midpoint of BC = ( (5 + 2)/2 , (8 + (-2))/2 )
= ( 7/2 , 6/2 )
= ( 7/2 , 3 )

So, the midpoint of AB is (1, 6) and the midpoint of BC is (7/2, 3).

Now, we can find the equation of the line containing the midsegment. The midsegment of a triangle is a line that connects the midpoints of two sides of the triangle. In this case, we want to find the equation of the line that connects the midpoint of AB and the midpoint of BC.

On this line, x will be the midpoint of AB (1) and z will be the midpoint of BC (7/2). Let's use the point-slope form of a line to find the equation:

Point-slope form: y - y₁ = m(x - x₁)

We can use the slope formula to find the slope (m) of the line connecting the midpoints:

Slope (m) = (y₂ - y₁) / (x₂ - x₁)

Using the coordinates of the midpoints (1, 6) and (7/2, 3):

Slope (m) = (3 - 6) / (7/2 - 1)
= (-3) / (7/2 - 2/2)
= (-3) / (5/2)
= (-3) * (2/5)
= -6/5

Now we can plug in the values into the point-slope form:

y - 6 = (-6/5)(x - 1)

To convert this equation into standard form (Ax + By = C), we can simplify:

Multiply both sides by 5 to eliminate the fraction:

5y - 30 = -6(x - 1)

Distribute -6 to the terms inside the parentheses:

5y - 30 = -6x + 6

Add 6x and 30 to both sides to isolate the variables on the left side:

6x + 5y = 36

Therefore, the equation of the line containing the midsegment in standard form, where x is the midpoint of AB and z is the midpoint of BC, is 6x + 5y = 36. The correct option is 4).