Post on 04-Jan-2016
We learned earlier that a segment bisector is any line, segment, or plane that intersects a segment at its midpoint. If a bisector is also perpendicular to the segment, it is called a perpendicular bisector.
Example 1:
a) Find the length of BC.
From the information in the diagram,
we know that is the perpendicular
bisector of .
CD
AB
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BC = AC Perpendicular Bisector TheoremBC = 8.5 Substitution
Example 1:
b) Find the length of XY.
Since and , is the
perpendicular bisector of by the converse
of the Perpendicular Bisector Theorem. By
the definition of segment bisector, .
Since 6, 6.
WX WZ WY XZ WY
XZ
XY ZY
ZY XY
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Example 1:
c) Find the length of PQ.
is the perpendicular bisector of .SQ PR�������������� �
PQ = RQ Perpendicular Bisector Theorem3x + 1 = 5x – 3 Substitution1 = 2x – 3 Subtract 3x from each side.4 = 2x Add 3 to each side.2 = x Divide each side by 2.
So, PQ = 3(2) + 1 = 7
When three or more lines intersect at a common point, the lines are called concurrent lines. The point where concurrent lines intersect is called the point of concurrency. A triangle has three sides, so it also has three perpendicular bisectors. These bisectors are concurrent lines. The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle.
Example 2: GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?
By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.
No, the circumcenter of an obtuse triangle is in the exterior of thetriangle.
Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle.
C
We learned earlier that an angle bisector divides an angle into two congruent angles. The angle bisector can be a line, segment, or ray.
Example 3:
b) Find mWYZ.
Since , , , is equidistant from the
sides of . By the Converse of the Angle Bisector Theorem,
bisects .
WX YX WZ YZ WX WZ W
XYZ
YW XYZ
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WYZ XYW Definition of angle bisectormWYZ = mXYW Definition of congruent anglesmWYZ = 28° Substitution
Example 3:
c) Find the length of QS.
QS = SR Angle Bisector Theorem
4x – 1 = 3x + 2 Substitution
x – 1 = 2 Subtract 3x from each side.
x = 3 Add 1 to each side.
So, QS = 4(3) – 1 or 11.
The angle bisectors of a triangle are concurrent, and their point of concurrency is called the incenter of a triangle.
Example 4:
a) Find ST if S is the incenter of ΔMNP.
By the Incenter Theorem, since S is equidistant from the sides of ΔMNP, ST = SU.
Find ST by using the Pythagorean Theorem.
a2 + b2 = c2 Pythagorean Theorem
82 + SU2 = 102 Substitution
64 + SU2 = 100 82 = 64, 102 = 100
Since length cannot be negative, use only the positive square root, 6. Since ST = SU, ST = 6.
SU2 = 36 Subtract 64 from each side.
SU = ±6 Take the square root of each side.
Example 4:
b) Find mSPU if S is the incenter of ΔMNP.
Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62°. Likewise, mTNU = 2mSNU, so mTNU = 2(28) or 56°.
mUPR + mRMT + mTNU = 180° Triangle Angle Sum Theorem
mUPR + 62° + 56° = 180° SubstitutionmUPR + 118° = 180° Simplify.
mUPR = 62° Subtract 118° from each side.