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Transcript
Aim #10: How do we construct a circumcenter and an incenter? CC Geometry H Do Now: Construct the perpendicular bisector of each side. 360° 122 B A C When three or more lines intersect in a single point, they are concurrent, and the point of intersection is the point of concurrency. In the diagram above, • label D, E, and F, the midpoints of AB, BC and AC, respectively. • mark the right angles • mark the congruent segments. • label P, the circumcenter 1. P is equidistant from A and B since it lies on the perpendicular bisector of ____. 2. P is equidistant from B and C since it lies on the perpendicular bisector of ____. 3. P is equidistant from A and C since it lies on the perpendicular bisector of ____. 4. Therefore, ____ = ____ = ____ 5. Draw circle P with radius PA. The intersection of the three perpendicular bisectors of a triangle is the circumcenter, the center of the circumscribed circle drawn above. The circumscribed circle is a circle that passes through every vertex of a triangle. The circumcenter can be located inside, outside or on the triangle. Using your compass and straightedge, construct the circumcenter Q of ΔABC. Then draw the circumscribed circle. ∆ABC is a(n) __________ triangle, therefore the circumcenter is located __________ of the circle. A C B 138° 118 How can you find the center of a circle, by construction, if the center is not shown? The intersection of the threeangle bisectors of a triangle is theincenter, the center of the inscribed circle. The inscribed circle is inside the triangle, and touches each of the sides in exactly one point. Q of ΔABC. 1. Using your compass and straightedge, construct the incenter 43 B A 138° C 2. State your steps for the above construction. 3. Any point on an angle bisector is _____________________ from the sides forming the angle. Since Q is on the angle bisector of ABC, it is _________________ from ___ and ___. Similarly, since Q is on the angle bisector of BCA, it is ___________________ from ____ and ____. Therefore, Q must also be equidistant from ___ and ___, since it lies on the bisector of BAC. Point Q is the point of ___________________ of the three angle bisectors. • The point of concurrency of the angle bisectors must be inside the triangle. • The point of concurrency of the angle bisectors is equidistant from the three sides when the perpendicular segment is drawn. 4. Point A is the _____________________ of ΔJKL. Point B is the ____________________of ΔRST. 5. Construct the incenter and the inscribed circle. 201 136° Let's Sum it Up!! The incenter of a triangle is the center of the circle that is inscribed in that triangle. Website to review construction of incenter: http://www.mathsisfun.com/geometry/constructtriangleinscribe.html The circumcenter of a triangle is the center of the circle that circumscribes that triangle. Website to review construction of circumcenter: http://www.mathsisfun.com/geometry/constructtrianglecircum.html Name ______________________ Date ________________ 1a) Given line segment AB, use a compass and straightedge to construct the set of points that are equidistant from A and B. CC Geometry H HW #10 b) Using a compass and straightedge, construct the set of points equidistant from the sides of the angle. A B 2a) Construct the circumcenter P of ΔXYZ. Z b) Name a fact about point P. c) Construct circumscribed circle P. X Y 3a) Construct the incenter I of ΔQRS. b) Name a fact about point I. c) Construct inscribed circle I. Q R S OVER Review: 1. For each of the following, construct a line perpendicular to segment AB that goes through point P. A P A P B B 2. Find x. Then classify the triangle by itsangles and sides. a) x = __________ 2x b) x = __________ 960 _____________ _____________ 5x 540 _____________ 3. Find the value of x that would make a ll b. x 480 a b _____________ 4x + 20 5x - 2 2 0 4. The measures of the base angles of an isosceles triangle are (x- 8x + 10) and 2 0 (2x - 5x) . Find the measure of the vertex angle.