How to Split a Triangle
We have learned that a segment bisector is any segment, line or plane that intersects a segment at its midpoint. Let’s take a look at a specific segment bisector.
The perpendicular bisector is a bisector that is also perpendicular to the segment. The perpendicular bisector is equidistant (the same distance) from the endpoints of the segment.
We have learned that a segment bisector is any segment, line or plane that intersects a segment at its midpoint. Let’s take a look at a specific segment bisector.
The perpendicular bisector is a bisector that is also perpendicular to the segment. The perpendicular bisector is equidistant (the same distance) from the endpoints of the segment.
\(\overleftrightarrow{CD}\) is the perpendicular bisector of \(\overline{AB}\)
Construct \(\overline{CE}\) and \(\overline{EU}\). Measure their lengths. What do you notice? Create another point, \(S\), on \(\overleftrightarrow{EL}\). Does the same property hold?
Example 1:
Determine if point \(P\) is on the perpendicular bisector using the given information in each diagram.
Determine if point \(P\) is on the perpendicular bisector using the given information in each diagram.
Solution:
a) Yes, we are given \(\overline{MQ}\cong\overline{OQ}\) and \(\overline{PM}\cong\overline{PO}\). That means that \(P\) is equidistant from the endpoints of a segment so it has to be on the perpendicular bisector.
b) No, we are only given one set of sides marked congruent, \(\overline{QS}\cong\overline{RS}\). This is not enough information to determine if \(P\) is on the perpendicular bisector.
Example 2:
Find each measure.
a) Yes, we are given \(\overline{MQ}\cong\overline{OQ}\) and \(\overline{PM}\cong\overline{PO}\). That means that \(P\) is equidistant from the endpoints of a segment so it has to be on the perpendicular bisector.
b) No, we are only given one set of sides marked congruent, \(\overline{QS}\cong\overline{RS}\). This is not enough information to determine if \(P\) is on the perpendicular bisector.
Example 2:
Find each measure.
a) \(AB = 7.6\)
Since \(A\) is on the perpendicular bisector, it is equidistant from \(B\) and \(C\). |
b) \(YW = 22\)
Since \(Z\) is the midpoint of \(\overline{WY}\), \(WZ = ZY\). So \(YW = 2(11)\). |
Example 3:
Find each measure.
Find each measure.
a)
\(\begin{align*}\\ JK &= LK\\ 4x+ 7 &= 6x - 9\\ -2x + 7 &= -9\\ -2x &= -16\\ x &= 8 \end{align*}\) \(JK = 4(8) + 7 = 32 + 7 = 39\) |
b)
\(\begin{align*}\\ GF &= EF\\ 7x + 5 &= 11x - 3\\ 5 &= 4x - 3\\ 8 &= 4x\\ 2 &= x \end{align*}\) \(GE = 2(GF) = 2(7(2) + 5) = 2(14+ 5) = 2(19) = 38\) |
An angle bisector is a ray that divides an angle into two congruent adjacent angles.
This GeoGebra applet shows that any point on the angle bisector of an angle is equidistant from the sides of the angle - click below:
This GeoGebra applet shows that any point on the angle bisector of an angle is equidistant from the sides of the angle - click below:
Example 4:
Solution:
a) Yes because \(\angle WXZ\cong\angle YXZ\), \(\overline{XW}\perp\overline{WZ}\), and \(\overline{XY}\perp\overline{ZY}\) so \(WZ = YZ\).
b) No because we don't know if \(\angle MNP\cong\angle ONP\).
Example 5:
Find each measure.
a) Yes because \(\angle WXZ\cong\angle YXZ\), \(\overline{XW}\perp\overline{WZ}\), and \(\overline{XY}\perp\overline{ZY}\) so \(WZ = YZ\).
b) No because we don't know if \(\angle MNP\cong\angle ONP\).
Example 5:
Find each measure.
a) \(IJ = 11.5\) since \(J\) is on the angle bisector
\(IJ = KJ\). |
b) \(m\angle ABD = 21^{\circ}\) since \(D\) is equidistant from the sides of the
angle, \(\angle CBD\cong\angle ABD\). |
Example 6:
Find each measure.
Find each measure.
a)
\(\begin{align*}\\ m\angle STV &= m\angle ETV\\ 7x - 14 &= 2x + 6\\ 5x - 14 &= 6\\ 5x &= 20\\ x &= 4 \end{align*}\) \(m\angle STV = 7(4) - 14 = 28 - 14 = 14^{\circ}\) |
b)
\(\begin{align*}\\ DC &= BC\\ 4x - 6 &= 8x - 18\\ 4x &= 8x - 12\\ -4x &= -12\\ x &= 3 \end{align*}\) \(DC = 4(3) - 6 = 12 - 6 = 6\) units |
A median of a triangle is a segment that goes from the vertex to the midpoint of the opposite side.
\(\overline{AD}\) is a median of \(\triangle ABC\)
An altitude of a triangle is a perpendicular segment that goes from the vertex to the opposite side or to the line that contains the opposite side. An altitude can lie in the interior, exterior or on the side of the triangle.
\(\overline{BD}\) is an altitude from \(B\) to \(\overline{AC}\)
|
\(\overline{BA}\) is an altitude from \(B\) to \(\overline{AC}\)
|
Example 7:
Use the given information to determine whether is a perpendicular bisector, angle bisector, median and/or altitude of \(\triangle NOL\).
Use the given information to determine whether is a perpendicular bisector, angle bisector, median and/or altitude of \(\triangle NOL\).
Solution:
a) altitude
b) median
c) angle bisector
d) all
a) altitude
b) median
c) angle bisector
d) all
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