10) Think about the formulas for distance and slope. In the diagram, \(M, N,\) and \(P\) are midpoints of \(\overline{AB}\), \(\overline{AC}\) , and \(\overline{BC}\) respectfully.
a) Find \(AB\) and \(NP\) and compare b) Find \(AC\) and \(MP\) and compare c) Find \(BC\) and \(MN\) and compare d) Find the slopes (rise to run ratio) of \(\overline{AB}\) and \(\overline{PN}\). Compare. What does that say about segments? e) Similar to part d of this problem, find: Slopes of \(\overline{BC}\) and \(\overline{MN}\) Slopes of \(\overline{CA}\) and \(\overline{PM}\) |
14) Write equations for two of the lines for that contain the midsegments.
17) Coordinate Proof of the Midsegment Theorem: \(\triangle HJK\) is a triangle of indiscriminate length. For example, the length of \(HJ\) is “\(2a\)” units long. We don’t know the length of “\(a\)”, but we are not concerned about that. We are going to show that a segment that joins the midpoints of \(\overline{JK}\) and \(\overline{JH}\) is both half the length and parallel to \(\overline{HK}\).
a. Find the ordered pair of the midpoint of \(\overline{JH}\) and call it point \(M\).
b. Find the ordered pair of the midpoint of \(\overline{JK}\) and call it point \(N\).
c. Find \(HK\) using the distance formula. Note, the distance will be written terms of “\(b\)” and “\(c\)”. That’s OK. Leave it that way.
d. Find \(MN\) in the same way you found \(HK\) from part c.
e. Compare \(HK\) and \(MN\).
f. Compare the slopes of \(\overline{HK}\) and \(\overline{MN}\). Use the slope formula for each. Again, it’s OK to leave the answer in terms of “\(b\)” and “\(c\)”.
g. Write a concluding statement regarding midsegment \(\overline{MN}\) and how it compares to the third side \(\overline{HK}\).
a. Find the ordered pair of the midpoint of \(\overline{JH}\) and call it point \(M\).
b. Find the ordered pair of the midpoint of \(\overline{JK}\) and call it point \(N\).
c. Find \(HK\) using the distance formula. Note, the distance will be written terms of “\(b\)” and “\(c\)”. That’s OK. Leave it that way.
d. Find \(MN\) in the same way you found \(HK\) from part c.
e. Compare \(HK\) and \(MN\).
f. Compare the slopes of \(\overline{HK}\) and \(\overline{MN}\). Use the slope formula for each. Again, it’s OK to leave the answer in terms of “\(b\)” and “\(c\)”.
g. Write a concluding statement regarding midsegment \(\overline{MN}\) and how it compares to the third side \(\overline{HK}\).
Review
18) Write an equation of a line perpendicular to \(-3x - 11y = -7\) through the point \((-5, 13)\)
18) Write an equation of a line perpendicular to \(-3x - 11y = -7\) through the point \((-5, 13)\)