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Constructions Involving Circles Section 7.4

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Definitions Concurrent: When three or more lines meet at a single point Circumcenter of a Triangle: The point where the 3 perpendicular bisectors of a triangle intersect. Centroid of a Triangle: The point where the 3 medians of a triangle intersect. Incenter of a Triangle: The point where the 3 angle bisectors of a triangle intersect. Orthocenter of a Triangle: The point where the 3 altitudes of a triangle intersect.

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Constructing the Circumscribed Circle of a Triangle 1.Construct a triangle 2.Construct the perpendicular bisector to any 2 sides 3.The intersection of the two perpendicular bisectors is the circumcenter of the triangle. If you place your compass tip at the intersection and extend it to a vertex on the triangle, you can construct a circumscribed circle. 4.This point, the circumcenter, is equidistant from what?

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Constructing the Inscribed Circle of a Triangle 1.Construct a triangle. 2.Construct any two angle bisectors of the triangle. 3.The intersection of the angle bisectors is the incenter. 4.The incenter is equidistant from what?

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Constructing the Centroid of a Triangle 1.Construct a triangle. 2.Construct the midpoints of any two sides and draw the medians. Use the perpendicular bisector construction to find the two midpoints. A median connects a midpoint and the opposite vertex of a triangle. 3.The intersection of the two medians is the centroid. 4.Whats the importance of the centroid of a triangle?

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Centroid Theorem In triangle ABC, if G is the centroid, then: CG = 2/3 CM C GM C = 1/3 CM C BG = 2/3 BM B GM B = 1/3 BM B AG = 2/3 AM A GM A = 1/3 AM A

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Constructing the Orthocenter of a Triangle 1.Construct a triangle. 2.Construct two altitudes. Use the construction perpendicular to a line through a given point not on the line You might need to extend the sides of the triangle. 3.The intersection of the altitudes is called the orthocenter.

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Constructing a Tangent to a Circle at a Point on the Circle 1.Draw the radius to the given point, P. 2.Construct a perpendicular to the radius through point P. Use the construction a perpendicular to a line through a point on the line.

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Construct a Tangent to a Circle from a Point Outside the Circle 1.Consider circle with center O and a point P outside the circle. 2.Construct the midpoint M of OP. 3.Construct the circle with center M and radius MP and MO. 4.Let Q be the point where the two circles intersect. 5.PQ will be tangent to circle O.

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Definitions Common Internal Tangents to 2 Circles: If two circles do not intersect, then a tangent to the two circles that crosses the segment connecting the two centers of the circles is called a common internal tangent. Common External Tangent to 2 Circles: If two circles do not intersect, then a tangent to the two circles that does not cross the segment connecting the two centers of the circles is called a common external tangent.

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Construct a Common Internal Tangent to Two Circles 1.Consider circles O and P. 2.Draw OP, giving two points of intersection Q and R. 3.Using O as center, construct a circle whose radius is OQ + PR. 4.Construct a tangent line from P to the new circle S using the previous construction. 5.Draw OS, calling T the point where OS intersects the original circle with center O. 6.Construct a line l, through T, parallel to SP. Line l is the common internal tangent to the original two circles.

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