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Triangles: Points of Concurrency. MM1G3 e. Investigate Points of Concurrency. http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html. Circumcenter. Perpendicular Bisectors and Circumcenters Examples.

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## Triangles: Points of Concurrency

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**Investigate Points of Concurrency**• http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html**A perpendicular bisector of a triangle is a line or line**segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisectorof the side. A B C D**Example 1:**A F E C B D**Example 2:**M Q N P**Since a triangle has three sides, it will have three**perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter. D G is the circumcenter of ∆DEF. Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices. G E F**The circumcenter will be located inside an acute triangle**(fig.1),outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters. fig. 1 fig. 3 fig. 2**Example 3: A company plans to build a new distribution**center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?**The circumcenter is equidistant to all three vertices of a**triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same. C**In Summary**• The circumcenter is the point where the three perpendicular bisectors of a triangle intersect. • The circumcenter can be inside, outside, or on the triangle. • The circumcenter is equidistant from the vertices of the triangle**Circumcenter**• Exploration • Construction**A median of a triangle is a line segment that contains the**vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.**Since a triangle has three sides, it will have three**medians. These medians will meet at a common point – the centroid.**The centroid is always located inside the triangle.**Acute triangle**Q**The distance from any vertex to the centroid is 2/3 the length of the median. E F G S R D**Example 1: G is the centroid of triangle QRS. QG = 10 GF =**3. Find QD and SF. Q E F G S R D**Example 3: G is the centroid of triangle DEF. FG = 15, ES =**21, QG = 5 Determine FR, EG and GD E Q 5 G 15 F R 21 S D**Notice that the distance from any vertex to the centroid is**2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median. So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ M V N Q U T P**The centroid is also known as the balancing point (center of**gravity) of a triangle.**In Summary**• A median is a line segment from the a vertex of a triangle to the midpoint of the opposite side. • The distance from the vertex to the centroid is 2/3 the length of the median. • The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid. • Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.**Centroids**• Investigate • Construction**An angle bisector of a triangle is a segment that shares a**common endpoint with an angle and divides the angle into two equal parts.**Example 1: Determine any angle bisectors oftriangle ABC.**A G F E C B D**Since a triangle has three angles, it will have three angle**bisectors. These angle bisectors will meet at a common point – the incenter. X M Z Y**The incenter is always located inside the triangle.**incenter Acute triangle Obtuse triangle Right triangle**The incenter is equidistant to the sides of the triangle.**x x**Example 2: L is the incenter of triangle ABC. Which**segments are congruent? A D B L E F C**Example 3: Given P is the incenter of triangle RST. PN = 10**and MT = 12, find PM and PT. 12 10 Not drawn to scale**In Summary**• The incenter is the point of intersection of the three angle bisectors of a triangle. • The incenter is equidistant to all three sides of the triangle.**Incenter**• Investigate • Construction**Try These:**K C

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