5-4 Bisectors in , Medians, and Altitudes

Do Now Lesson Presentation Exit Ticket 5-4 Bisectors in Triangles, Medians, and Altitudes Connect to Mathematical Ideas (1)(F) By the end of today’s lesson, SWBAT . Prove and apply properties of bisectors of a .

. Prove and apply properties of bisectors of a triangle. 5-4 Bisectors in Triangles, Medians, and Altitudes

Vocabulary concurrent of concurrency circumcenter of a triangle circumscribed of a triangle inscribed 5-4 Bisectors in Triangles, Medians, and Altitudes

Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property. 5-4 Bisectors in Triangles, Medians, and Altitudes Helpful Hint The perpendicular bisector of a side of a triangle does not always pass through the opposite . 5-4 Bisectors in Triangles, Medians, and Altitudes 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. 5-4 Bisectors in Triangles, Medians, and Altitudes

The circumcenter can be inside the triangle, outside the triangle, or on the triangle. 5-4 Bisectors in Triangles, Medians, and Altitudes

The circumcenter of ΔABC is the center of its circumscribed . A circle that contains all the vertices of a polygon is circumscribed about the polygon. 5-4 Bisectors in Triangles, Medians, and Altitudes Example 1: 푫푮, 푬푮, and 푭푮 are the perpendicular bisectors of ∆ABC. Find GC. G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ∆ABC. GC = GB Circumcenter Thm.

GC = 13.4 Substitute 13.4 for GB. 5-4 Bisectors in Triangles, Medians, and Altitudes Example 2: Finding the Circumcenter of a Triangle Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).

Step 1 Graph the triangle.

Step 2 Find equations for two perpendicular bisectors. x = 5 y = 3

Step 3 Find the intersection of the two equations. The lines x = 5 and y = 3 intersect at (5, 3), the circumcenter of ∆HJK. 5-4 Bisectors in Triangles, Medians, and Altitudes

A triangle has three , so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle . 5-4 Bisectors in Triangles, Medians, and Altitudes

Remember! The distance between a point and a is the of the perpendicular segment from the point to the line. 5-4 Bisectors in Triangles, Medians, and Altitudes Unlike the circumcenter, the incenter is always inside the triangle. 5-4 Bisectors in Triangles, Medians, and Altitudes

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. 5-4 Bisectors in Triangles, Medians, and Altitudes Example 3: Identifying and Using the Incenter of ∆ Algebra GE = 2x – 7 and GF = x + 4. What is GD ? G is the incenter of ∆ABC because it is the point of concurrency of the angle bisectors. By the Concurrency of Angle Bisectors Theorem, the distances from the incenter to the three sides of the triangle are equal, so GE = GF = GD. Use this relationship to find x. 2푥 − 7 = 푥 + 4 GE = GF 푥 − 7 = 4 Subtract x from each side

푥 = 11 Add 7 to each side ⇒ 푮푭 = 풙 + ퟒ 퐺퐹 = 11 + 4 ⇔ 퐺퐹 = 15 ∴ 퐺퐸 = 퐺퐹 = 퐺퐷 ⇔ 15 5-4 Bisectors in Triangles, Medians, and Altitudes

A of a triangle is a segment whose endpoints are a vertex of the triangle and the of the opposite side.

Every triangle has three medians, and the medians are concurrent. 5-4 Bisectors in Triangles, Medians, and Altitudes The point of concurrency of the medians of a triangle is the of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance. 5-4 Bisectors in Triangles, Medians, and Altitudes

Example 4: Using the Centroid to Find Segment In ∆LMN, RL = 21 and SQ =4. Find LS.

Centroid Theorem.

Substitute 21 for RL.

LS = 14 Simplify. 5-4 Bisectors in Triangles, Medians, and Altitudes

Example 5: Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find NQ.

Centroid Thm.

NS + SQ = NQ Seg. Add. Post.

Substitute NQ for NS.

Subtract NQ from both sides.

Substitute 4 for SQ.

12 = NQ Multiply both sides by 3. 5-4 Bisectors in Triangles, Medians, and Altitudes

An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle. 5-4 Bisectors in Triangles, Medians, and Altitudes In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle. 5-4 Bisectors in Triangles, Medians, and Altitudes

Helpful Hint The height of a triangle is the length of an altitude. 5-4 Bisectors in Triangles, Medians, and Altitudes 5-4 Bisectors in Triangles, Medians, and Altitudes

Example 5: Finding the Orthocenter ∆ABC has vertices A(1,3), B(2,7), and C(6,3). What are the coordinates of the orthocenter of ∆ABC ? Step 1 Graph the triangle.

Step 2 Find an equation of the line containing the altitude from B to 푨푪. x = 2 B Step 3 Find an equation of the line containing the altitude from A to 푩푪. y = x + 2 A C Step 4 Find the orthocenter by solving this x = 2 system of equations. y = x + 2 ∴ The coordinates of the orthocenter are (2, 4). 5-4 Bisectors in Triangles, Medians, and Altitudes Got It ? Solve With Your Partner Problem 1 Finding the length of the median. a. In the diagram at the right, ZA = 9. What is the length of 풁푪 ? 13.5 b. What is the ratio of ZA to AC ? Explain. 2:1 5-4 Bisectors in Triangles, Medians, and Altitudes Got It ? Solve With Your Partner Problem 2 Finding the circumcenter. What are the coordinates of the circumcenter of the triangle with vertices A(2,7), B(10,7), and C(10,3) ?

∴ The coordinates of the circumcenter = (6,5) 5-4 Bisectors in Triangles, Medians, and Altitudes Got It ? Solve With Your Partner Problem 3 Finding the orthocenter. ∆DEF with vertices D(1,2), E(1,6), and F(4,2). What are the coordinates of the orthocenter of ∆DEF ?

E ∴ The coordinates of the orthocenter = (1,2) D F 5-4 Bisectors in Triangles, Medians, and Altitudes Closure: Communicate Mathematical Ideas (1)(G) .What are the properties of the circumcenter of a triangle? It is equidistant from the vertices of the triangle and contained in the perpendicular bisectors of the sides of the triangle. . What are the properties of the incenter of a triangle? It is equidistant from the sides of the triangle and contained in the bisectors of the angles of the triangle. .Where do the medians of a triangle intersect? Does the point have any special characteristics? The medians intersect at a point called the centroid. It is locate ⅔ of the way from a vertex to its opposite side. .Where do the altitudes of a triangle intersect? The altitudes intersect at a point called the orthocenter. 5-4 Bisectors in Triangles, Medians, and Altitudes Exit Ticket: Use the figure for Items 1–3. In ∆ABC, AE = 12, DG = 7, and BG = 9. Find each length. 1. AG 8 2. GC 14 3. GF 13.5 For Items 4 and 5, use ∆MNP with vertices M (–4, –2), N (6, –2), and P (–2, 10). Find the coordinates of each point.

4. the centroid (0, 2) 5. the orthocenter