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Warm- up Type 2 writing and Construction • Write your own definition and draw a picture of the following: • Angle Bisector • Perpendicular Bisector • Draw an acute triangle. Make the sides at least three inches in length. • Construct the perpendicular bisector of one of the sides of the triangle. 5.3: Concurrent Lines, Medians and Altitudes, Angle Bisectors, Perpendicular Bisectors Objectives: •To identify properties of perpendicular bisectors and angle bisectors •To identify properties of medians and altitudes of triangles Centroids • Draw a large acute scalene triangle on your index card. • Measure and mark the midpoint of each side of the triangle. • Draw a line from each midpoint to the opposite vertex. • These three lines should meet at one point! • Cut your triangle out. • Try to balance the triangle by putting your pencil tip on the point you located. Median of a triangle – a segment whose endpoints are the vertex of the triangle and the midpoint of the opposite side. Centroid – The point of concurrency of the three medians of a triangle. The centroid is always inside the triangle. It is the center of gravity of the triangle. Theorem 5-7 The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. (The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.) Median of a Triangle A segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side G D BG = 9.21 cm A AB = 8.75 cm DB = 3.44 cm B BC = 4.37 cm BE = 6.88 cm FB = 4.60 cm F E C CENTROID: Point of concurrency of medians of a triangle The medians of a triangle intersect at a point that is 2/3 the distance from the vertex to the midpoint of the opposite side. G is the centroid AG = 2 3 BG = 2 CG= 2 3 3 AD BF CE O is the centroid of triangle ABC. CE = 11 FO = 10 CO = 8 1. Find EB. 2. Find OD. 3. Find FB. 4. Find CD. 5. If AO = 6, find OE. Centroid – The point of concurrency of the medians of a triangle. A centroid is always inside the triangle. A centroid is the center of gravity in a triangle Concurrent Lines – three or more lines that intersect in the same point. Point of Concurrency – The point of intersection of three or more lines. Perpendicular Bisector of a Triangle – A line ray or segment that is perpendicular to a side of the triangle at the midpoint of the side. A B Circumcenter – the point of concurrency of the perpendicular bisectors of a triangle. - In an acute triangle the circumcenter is located inside the triangle. - In an obtuse triangle the circumcenter is located outside of the triangle. -In a right triangle the circumcenter is on the hypotenuse of the triangle. Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle If you were to draw a circle around a triangle, where each vertex of the triangle are points on the circle, the circle would be circumscribed about that triangle The circumcenter of the triangle is ALSO the center of the circle circumscribed about it Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (-8,0) and (0,6). 1. 2. 3. 4. Plot the points on a coordinate plane. Draw the triangle Draw the perpendicular bisectors of at least 2 sides. The circumcenter of this triangle will be the center of the circle. Perpendicular Bisectors Acute triangle: Circumcenter inside triangle Right triangle: Circumcenter lies ON the triangle Obtuse Triangle: Circumcenter is outside triangle Perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices of the triangle AB = 6.29 cm AC = 6.29 cm AD = 6.29 cm C A B D Properties of the Circumcenter -It is the center of a circle that passes through the vertices of the triangle -It is the point that is an equal distance from each vertex of the triangle. -In a right triangle it is on the triangle, in an acute triangle it is inside the triangle, in an obtuse triangle it is outside of the triangle. Theorem 5.5 –Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. Marshfield, Scituate and Hanover want to construct an indoor pool. They want the pool to be located an equal distance from the center of each town. Determine where the pool should be constructed. Hanover Scituate Marshfield Incenter of a triangle The point of concurrency of the three angle bisectors of a triangle Theorem 5-6 Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The incenter is always located inside the triangle. m B C A D The incenter is the center of a circle that you can inscribe within the triangle. When a circle is inscribed inside of a triangle the circle touches each side of the triangle at just one point. B C A D Altitude of a triangle – a perpendicular segment from the vertex to the opposite side or a line that contains the opposite side. Orthocenter – the point of concurrency of the three altitudes of a triangle. Altitude of a Triangle The perpendicular segment from the vertex to the line containing the opposite side Acute Triangle: Inside Right Triangle: Side Obtuse Triangle: Outside Orthocenter of a Triangle Point of concurrency for altitudes of a triangle Acute Triangle: Inside Right Triangle: Vertex Obtuse Triangle: Outside Midsegment of a Triangle – a segment that connects the midpoints of two sides of a triangle. Theorem 5-9 Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. Theorem 5-10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Theorem 5-11 If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Theorem 5-12 Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. Theorem 5-13 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.