Midsegments
A midsegment of a triangle is a segment which connects two midpoints. For example:
A midsegment of a triangle is a segment which connects two midpoints. For example:
In the Investigation you discovered two properties for midsegments.
Midsegment Properties
The midsegment joining the midpoints of two sides of the triangle is parallel to the third side of the triangle and half the length of the third side of the triangle.
The midsegment joining the midpoints of two sides of the triangle is parallel to the third side of the triangle and half the length of the third side of the triangle.
Here \(\overline{DE}\) is the midsegment of \(\triangle ABC\). So \(\overline{DE}\) is half the length of \(\overline{BC}\) (or \(\overline{BC}\) is twice as long as \(\overline{DE}\)). Also, \(\overline {DE} \parallel \overline{BC}\)
Solution:
a) three midsegments can be drawn in any triangle b) let's label the midpoints \(D\) and \(E\). Using the distance formula: \(DE = \sqrt{(-2 - -5)^2 + (7 - 3)^2}\) \( = \sqrt{(3)^2 + (4)^2}\) \( = \sqrt{9 + 16}\) \( = \sqrt{25} = 5\) c) Let's use point-slope form. The slope of the midsegment \(\overline{DE}\) is \(\Large\frac{4}{3}\)since we know that the slope of midsegment DE is 4/3 and a point on the line is point \(E (-2, 7)\). Note: We could have used point \(D\) or point \(E\), and so chose point \(E\) arbitrarily. \(y - 7 = \Large\frac{4}{3}\normalsize (x + 2)\) |
Solution:
Since \(\overline{DE}\) is a midsegment, points \(D\) and \(E\) are midpoints. Therefore \(\overline{AD}\cong\overline{DB}\) because midpoints create congruent segments. So \(x = 5\).
Since a midsegment is half the length of the side to which it is parallel, the length of \(\overline{DE}\) is half of \(9\), so \(y = 4.5\).
Since \(\overline{DE}\) is a midsegment, points \(D\) and \(E\) are midpoints. Therefore \(\overline{AD}\cong\overline{DB}\) because midpoints create congruent segments. So \(x = 5\).
Since a midsegment is half the length of the side to which it is parallel, the length of \(\overline{DE}\) is half of \(9\), so \(y = 4.5\).