1. Warm Up 1. Draw a triangle and construct the bisector of one angle.
2. JK is perpendicular to ML at its midpoint K. List the congruent segments.
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2. Since a triangle has three sides, it has three perpendicular bisectors
Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property.
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3. When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. The three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.
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4. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.
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5. The circumcenter of ΔABC is the center of its circumscribed circle
The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.
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6. Example 1: Using Properties of Perpendicular Bisectors
DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.
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7. MZ is a perpendicular bisector of ∆GHJ.
Example 1a
Use the diagram. Find GM.
MZ is a perpendicular bisector of ∆GHJ.
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8. KZ is a perpendicular bisector of ∆GHJ.
Example 1b
Use the diagram. Find GK.
KZ is a perpendicular bisector of ∆GHJ.
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9. Example 1c Use the diagram. Find JZ.
Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of
∆GHJ.
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10. A triangle has three angles, so it has three angle bisectors
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .
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11. Unlike the circumcenter, the incenter is always inside the triangle.
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12. The incenter is the center of the triangle’s inscribed circle
The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
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13. Example 3A: Using Properties of Angle Bisectors
MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.
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14. Example 3B: Using Properties of Angle Bisectors
MP and LP are angle bisectors of ∆LMN. Find mPMN.
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15. QX and RX are angle bisectors of ΔPQR. Find the distance from X to PQ.
Example 3a
QX and RX are angle bisectors of ΔPQR. Find the distance from X to PQ.
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16. QX and RX are angle bisectors of ∆PQR. Find mPQX.
Example 3b
QX and RX are angle bisectors of ∆PQR. Find mPQX.
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17. Every triangle has three medians, and the medians are concurrent.
A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Every triangle has three medians, and the medians are concurrent.
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18. The point of concurrency of the medians of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called
the center of gravity because it is the point where a triangular region will balance.
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19. Example 1A: Using the Centroid to Find Segment Lengths
In ∆LMN, RL = 21 and SQ =4.
Find LS.
Find NQ.
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20. Example 1a
In ∆JKL, ZW = 7, and LX = 8.1.
Find LZ.
Find KW.
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21. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.
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22. In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not
In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.
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23. 1. ED, FD, and GD are the perpendicular bisectors of ∆ABC. Find BD.
Lesson Review: Part I
1. ED, FD, and GD are the perpendicular bisectors of ∆ABC. Find BD.
2. JP, KP, and HP are angle bisectors of ∆HJK. Find the distance from P to HK.
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24. Lesson Review: Part II
3. Lee’s job requires him to travel to X, Y, and Z. Draw a sketch to show where he should buy a home so it is the same distance from all three
places.
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