Warm-Up2/24 1)What is one angle measure of a regular nonagon? 2)Find the missing angle

Announcements2/24 1)HW 5.4 Due Today 2)Homework 5.5 Due Monday 3)CM #26 on Monday 4)Unit test on 5 th and 6 th which is a Thurs and Fri. 5)Bring compasses and straight edges for Wed- Mon.

5.5 Centers of Triangles

Review: How to construct a Perpendicular Bisector. How to construct an Angle Bisector. How to find the median of a line segment.

Warm-Up2/25 1)Draw a line segment 3 inches lone. 2)Construct a Perpendicular Bisector of this line segment

Cont of 5.5 Centers of Triangles

Theorem 5.13 Two lines intersected by a transversal are parallel if and only if the consecutive interior angles are congruent. Theorem 5.14 Circumcenter Theorem. The perpendicular bisectors of the sides of any triangle are concurrent at the circumcenter, which is equidistant from each vertex of the triangle. Meaning: the Circumcenter would be the center of a circle that would circumscribe a triangle. Where it can go: The circumcenter can lie in the exterior of the triangle.

Steps to find the circumcenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Make 2)Make two small arcs on either side of the line apprxmtly where the perpendicular bisector would go. 3)Connect 3)Connect the intersections with a line. 4)Repeat 4)Repeat steps 1-3 for the other two vertices. 5)The 5)The point of intersection of the three lines is the circumcenter.

NOW YOU DO IT. Use one full piece of paper. Draw a triangle with your straight edge. Steps to find the circumcenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Make 2)Make two small arcs on either side of the line apprxmtly where the perpendicular bisector would go. 3)Connect 3)Connect the intersections with a line. 4)Repeat 4)Repeat steps 1-3 for the other two vertices. 5)The 5)The point of intersection of the three lines is the circumcenter.

Theorem 5.15 Incenter Theorem. The angle bisectors of the angles of a triangle are concurrent at the incenter, which is equidistant from the sides of the triangle. Meaning: The incenter would be the center of a circle that would inscribe a circle. Where it can lie: The incenter of the triangle will always lie in the interior of the triangle.

Steps to find the incenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Intersect 2)Intersect the two sides that contain the vertex with two small little arcs. 3)Now 3)Now put the point of your compass on the intersection of the arc and the side and make a small arcs apprxmtly where the angle bisector would go. 4)Repeat 4)Repeat step 3 from the other intersection so that the arcs intersect. 5)Connect 5)Connect the intersections with a line. 6)Repeat 6)Repeat steps 1-5 for the other two vertices. 7)The 7)The point of intersection of the three lines is the incenter

NOW YOU DO IT. Use one full piece of paper. Draw a triangle with your straight edge. Steps to find the incenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Intersect 2)Intersect the two sides that contain the vertex with two small little arcs. 3)Now 3)Now put the point of your compass on the intersection of the arc and the side and make a small arcs apprxmtly where the angle bisector would go. 4)Repeat 4)Repeat step 3 from the other intersection so that the arcs intersect. 5)Connect 5)Connect the intersections with a line. 6)Repeat 6)Repeat steps 1-5 for the other two vertices. 7)The 7)The point of intersection of the three lines is the incenter

Warm-Up2/26 1)What is the meaning of the incenter and circumcenter?

Cont of 5.5 Centers of Triangles

An altitude of a triangle triangle is a segment that extends from a vertex and is perpendicular to the opposite side. Theorem 5.16 orthocenter Theorem. The lines that contain the three altitudes are concurrent at the orthocenter. Meaning: The orthocenter only has meaning in higher mathematics. Where it can lie: The orthocenter can lie in the exterior of the triangle

Steps to find the orthocenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Intersect 2)Intersect the side opposite the vertex with two small little arcs. NOTE: If you need to extend a side use your straight edge to extend the side. 3)Now 3)Now put the point of your compass on the intersection of the arc and the side and make two small arcs apprxmtly where the bisector would go. 4)Repeat 4)Repeat step 3 from the other intersection so that the arcs intersect. 5)Connect 5)Connect the intersections with a line. 6)Repeat 6)Repeat steps 1-5 for the other two vertices. 7)The 7)The point of intersection of the three lines is the ortocenter.

NOW YOU DO IT. Use one full piece of paper. Draw a triangle with your straight edge. Steps to find the orthocenter of a triangle: 1)Put 1)Put point of compass on a vertex of the triangle. 2)Intersect 2)Intersect the side opposite the vertex with two small little arcs. NOTE: If you need to extend a side use your straight edge to extend the side. 3)Now 3)Now put the point of your compass on the intersection of the arc and the side and make two small arcs apprxmtly where the bisector would go. 4)Repeat 4)Repeat step 3 from the other intersection so that the arcs intersect. 5)Connect 5)Connect the intersections with a line. 6)Repeat 6)Repeat steps 1-5 for the other two vertices. 7)The 7)The point of intersection of the three lines is the ortocenter.

Warm-Up2/27 1)Draw a triangle 2)Construct one altitude of the triangle

Cont of 5.5 Centers of Triangles

A median of a triangle triangle is a segment extending from a vertex to the midpoint of the opposite side. Theorem 5.17 Centroid Theorem. The three medians of a triangle are concurrent at the centroid. Meaning: The centroid is the center of mass. Where it can lie: The centroid will always lie on the interior of the triangle.

Steps to find the centroid of a triangle: 1)Find 1)Find the midpoint of each side of the triangle. a)Put a)Put point of compass on the vertex of a line segment and make a small arc on either side of the line apprxmtly where the segment bisector would go. b)Repeat b)Repeat a) for the other vertex. c)Make c)Make a mark on the side of the triangle to mark the midpoint. 2)Connect 2)Connect the vertex to the midpoint of the side opposite the vertex. 3)Repeat 3)Repeat steps 1-2 for the other two vertices. 4)The 4)The point of intersection of the three lines is the centroid.

NOW YOU DO IT. Use one full piece of paper. Draw a triangle with your straight edge. Steps to find the centroid of a triangle: 1)Find 1)Find the midpoint of each side of the triangle. a)Put a)Put point of compass on the vertex of a line segment and make a small arc on either side of the line apprxmtly where the segment bisector would go. b)Repeat b)Repeat a) for the other vertex. c)Make c)Make a mark on the side of the triangle to mark the midpoint. 2)Connect 2)Connect the vertex to the midpoint of the side opposite the vertex. 3)Repeat 3)Repeat steps 1-2 for the other two vertices. 4)The 4)The point of intersection of the three lines is the centroid.