Geometry Bisector of Triangles CONFIDENTIAL

Warm up Determine whether each point is on the perpendicular bisector of the segment with endpoints S(0,8) and T(4,0). 1) X(0,3) 2) Y(-4,1) 3) Z(-8,-2) yes no CONFIDENTIAL

Bisector of Triangles Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property. CONFIDENTIAL

Draw a large scalene acute triangle ABC on a piece of patty paper. 1 2 3 B A B P C A C A C Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. CONFIDENTIAL

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. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. CONFIDENTIAL

Circumcenter Theorem Theorem 2.1 B The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC P A B C CONFIDENTIAL

The circumcenter can be inside the triangle, Out the triangle, or on the triangle. P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

The circumcenter of ∆ABC is the center of its circumscribed circle The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. B P A C CONFIDENTIAL

Circumcenter Theorem A l n P B C m CONFIDENTIAL

Using Properties of Perpendicular Bisector H J G K L 19.9 Z M 18.6 14.5 9.5 Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ. HZ = GZ Circumcenter Thm. HZ = 19.9 Substitute 19.9 for GZ. CONFIDENTIAL

Use the diagram. Find each length. Now you try! Use the diagram. Find each length. a) GM b) GK c) JZ H J G K L 19.9 Z M 18.6 14.5 9.5 1a) 14.5 1b) 18.6 1c) 19.9 CONFIDENTIAL

Finding the Circumcenter of a Triangle. Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0). Step 1 Graph the triangle. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

Step 2 Find equation for two perpendicular bisectors. Since Two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of RO is x = -3, and the perpendicular bisector of OS is y =2. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

Step 3 Find the intersection of the two equations. The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y X = -4 S Y = 2 (-3,2) x R 4 O CONFIDENTIAL

2) Find the circumcenter of ∆GOH with vertices Now you try! 2) Find the circumcenter of ∆GOH with vertices G(0,-9), O(0,0), and H(8,0). 2) 29 CONFIDENTIAL

Theorem 2.2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ B Z Y P C A X CONFIDENTIAL

Unlike the circumcenter, the incenter is always inside the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

The incenter is the center of the triangle’s inscribed 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. P A B C CONFIDENTIAL

Using Properties of Angle Bisectors K A) The distance from V to KL 7.3 V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL. The distance from V to JK is 7.3. So the distance from V to KL is also 7.3 W V 106˚ J L 19˚ Next page: CONFIDENTIAL

JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K B) m/ Vkl JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K m/ kJL = 2m / VJL m/ kJL = 2(19˚) = 38˚ m/ kJL + m/ JLK + m/ JKL = 180˚ 38 + 106 + m/ JKL = 180˚ m/ JKL = 36˚ m/ Vkl = ½ m/ JKL m/ Vkl = ½ (36˚) = 18˚ Substitute the given values. Subtract 144˚ from both sides. 7.3 W KV is the bisector of m/ JKL V Substitute 36˚ for m/ JKL. 106˚ L 19˚ CONFIDENTIAL

Now you try! 3) QX and RX are angle bisectors ∆PQR. Find each measure. The distance from X to PQ m/ PQX Q X 3a) 14.5 3b) 18.6 R P 12˚ Y 19.2 CONFIDENTIAL

Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B Next page: CONFIDENTIAL

Trace the map. Draw the triangle formed by the viewing locations Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A B C F CONFIDENTIAL

Now you try! 4) A City plans to build a firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. Centerville Anverue King Boulevard Third Street CONFIDENTIAL

Now some practice problems for you! CONFIDENTIAL

Assessment SN, TN, and VN are the perpendicular bisectors of ∆PQR. Find each length. 1) NR 2) RV 3) TR 4) QN Q 3.95 S N T 5.64 4.03 P R 5.47 V CONFIDENTIAL

Find the circumcenter of a triangle with the given vertices. 5) O(0,0), K(0, 12), L(4,0) 6) A(-7,0), O(0,0),B(0,-10) CONFIDENTIAL

CF and EF are angle bisectors of ∆CDE. Find each measure. 7) The distance from F to CD 8) m/ FED C D E G F 42.1 54˚ 17˚ CONFIDENTIAL

9) The designer of the Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch. BEARS CONFIDENTIAL

Let’s review Bisector of Triangles Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property. CONFIDENTIAL

Draw a large scalene acute triangle ABC on a piece of patty paper. 1 2 3 B A B P C A C A C Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. CONFIDENTIAL

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. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. CONFIDENTIAL

Theorem 2.1 Circumcenter Theorem B The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC P C A CONFIDENTIAL

The circumcenter can be inside the triangle, Out the triangle, or on the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

The circumcenter of ∆ABC is the center of its circumscribed circle The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. B P A C CONFIDENTIAL

Circumcenter Theorem A l n P B C m CONFIDENTIAL

Using Properties of Perpendicular Bisector H 18.6 Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ. HZ = GZ Circumcenter Thm. HZ = 19.9 Substitute 19.9 for GZ. Z L K 9.5 19.9 G J M 14.5 CONFIDENTIAL

Finding the Circumcenter of a Triangle. Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0). Step 1 Graph the triangle. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

Step 2 Find equation for two perpendicular bisectors. Since Two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of RO is x = -3, and the perpendicular bisector of OS is y =2. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

Step 3 Find the intersection of the two equations. The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y X = -4 S Y = 2 (-3,2) x R 4 O CONFIDENTIAL

Theorem 2.2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ B Z Y P C A X CONFIDENTIAL

Unlike the circumcenter, the incenter is always inside the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

The incenter is the center of the triangle’s inscribed 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. B P A C CONFIDENTIAL

Using Properties of Angle Bisectors K A) The distance from V to KL 7.3 V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL. The distance from V to JK is 7.3. So the distance from V to KL is also 7.3 W V 106˚ J L 19˚ Next page: CONFIDENTIAL

JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K B) m/ Vkl JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K m/ kJL = 2m / VJL m/ kJL = 2(19˚) = 38˚ m/ kJL + m/ JLK + m/ JKL = 180˚ 38 + 106 + m/ JKL = 180˚ m/ JKL = 36˚ m/ Vkl = ½ m/ JKL m/ Vkl = ½ (36˚) = 18˚ Substitute the given values. Subtract 144˚ from both sides. 7.3 W KV is the bisector of m/ JKL V Substitute 36˚ for m/ JKL. 106˚ L 19˚ CONFIDENTIAL

Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B Next page: CONFIDENTIAL

Trace the map. Draw the triangle formed by the viewing locations Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A C F B CONFIDENTIAL

You did a great job today! CONFIDENTIAL