Math 1 Warm-ups 1-8-09 Fire stations are located at A and B. XY , which contains Havens Road, represents the perpendicular bisector of AB . A fire is reported at point X. Which fire station is closer to the fire? Explain.

What is the incenter of a triangle? Math 1 Warm-ups 1-9-09 What is the incenter of a triangle?

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.

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.

Math Support Warm-ups 1-8-09 True or false: An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.

Vocabulary concurrent point of concurrency circumcenter of a triangle incenter of a triangle Orthocenter Centroid

Point of concurrency 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.

Incenter of a circle The incenter is always inside the triangle. 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 . The incenter is always inside the triangle.

Incenter of a circle The incenter is always inside the triangle. 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 . The incenter is always inside the triangle.

Incenter of a 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. Point of concurrency is center of imaginary inscribed circle Incenter is equidistant from the sides PX = PY = PZ

Using Properties of Angle Bisectors Incenter Example: Using Properties of Angle Bisectors MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN. P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN. The distance from P to LM is 5. So the distance from P to MN is also 5.

Circumcenter center of a circle The three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

Point of concurrency is center of circumscribed circle Circumcenter of a triangle The three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. The circumcenter can be inside, outside, or on the triangle. Point of concurrency is center of circumscribed circle Circumcenter is equidistant from the vertices PA = PB = PC

Using Properties of Perpendicular Bisectors Circumcenter Example: Using Properties of Perpendicular Bisectors DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC. G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ∆ABC. GC = GB Circumcenter Thm. GC = 13.4 Substitute 13.4 for GB.

Centroid of a Triangle The point of concurrency of the median 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. AP to PY is 2:1 CP to PX is 2:1 ZP to PB is 2:1

Using the Centroid to Find Segment Lengths Centroid Example Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify.

The Orthocenter of a Circle The point of concurrency of the altitudes of a triangle is the orthocenter of the triangle.

The Orthocenter of a Circle 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.