5-4 Bisectors in Triangles, Medians, and Altitudes

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Lesson Presentation

5-4 Bisectors in Triangles, Medians, and Altitudes

SWBAT

Prove and apply properties of perpendicular

bisectors of a triangle.

Prove and apply properties of angle bisectors of a

triangle.

By the end of today’s lesson,

Connect to Mathematical Ideas (1)(F)

5-4 Bisectors in Triangles, Medians, and Altitudes

concurrent

point of concurrency

circumcenter of a triangle

circumscribed

incenter of a triangle

inscribed

Vocabulary

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

The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex.

Helpful Hint

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 circle. 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:

G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of ∆ABC.

𝑫𝑮, 𝑬𝑮, and 𝑭𝑮 are the perpendicular bisectors of ∆ABC. Find GC.

GC = GB

GC = 13.4

Circumcenter Thm.

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 angles, 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

The distance between a point and a line is the length of the perpendicular segment from thepoint to the line.

Remember!

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.

2𝑥 − 7 = 𝑥 + 4 GE = GF

𝑥 − 7 = 4 Subtract x from each side

𝑥 = 11 Add 7 to each side

⇒ 𝑮𝑭 = 𝒙 + 𝟒

𝐺𝐹 = 11 + 4 ⇔ 𝐺𝐹 = 15

∴ 𝐺𝐸 = 𝐺𝐹 = 𝐺𝐷 ⇔ 15

5-4 Bisectors in Triangles, Medians, and Altitudes

A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint 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 centroid 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 Lengths

In ∆LMN, RL = 21 and SQ =4. Find LS.

LS = 14

Centroid Theorem.

Substitute 21 for RL.

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.

12 = NQ

Substitute 4 for SQ.

Multiply both sides by 3.

Substitute NQ for NS.

Subtract NQ from both sides.

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

The height of a triangle is the length of an altitude.

Helpful Hint

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 𝑨𝑪.

A

B

C

x = 2

Step 3 Find an equation of the line containing

the altitude from A to 𝑩𝑪. y = x + 2

Step 4 Find the orthocenter by solving this

system of equations.x = 2y = 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 𝒁𝑪 ?

b. What is the ratio of ZA to AC ? Explain.

13.5

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 ?

D

E

F

∴ The coordinates of the

orthocenter = (1,2)

5-4 Bisectors in Triangles, Medians, and Altitudes

Closure: Communicate Mathematical Ideas (1)(G)

What are the properties of the circumcenter of a triangle?

What are the properties of the incenter of a triangle?

Where do the medians of a triangle intersect? Does the point have any special characteristics?

Where do the altitudes of a triangle intersect?

It is equidistant from the vertices of the triangle and contained in the perpendicular bisectors of the sides of the triangle.

It is equidistant from the sides of the triangle and contained in the bisectors of the angles of the triangle.

The medians intersect at a point called the centroid. It is locate ⅔ of the way from a vertex to its opposite side.

The altitudes intersect at a point called the orthocenter.

5-4 Bisectors in Triangles, Medians, and Altitudes

Use the figure for Items 1–3. In ∆ABC, AE = 12, DG = 7, and BG = 9. Find each length.

1. AG

2. GC

3. GF

8

14

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

5. the orthocenter

(0, 2)

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