5-1 Midsegments of Triangles

Quick Review

A midsegment of a triangle is a segment that connects the midpoints of two sides. A midsegment is parallel to the third side and is half as long.

Example

Algebra Find the value of x.

D E ¯ d e bar  is a midsegment because D and E are midpoints.

D E = 1 2 B C Δ  Midsegmet Theorem 2 x = 1 2 ( x + 12 ) Substitute . 4 x = x + 12 Simplify . 3 x = 12 Subtract  x  from each side . x = 4 Divide each side by  3. table with 5 rows and 3 columns , row1 column 1 , d e , column 2 equals , 1 half b c , column 3 cap delta . cap midsegmetcap theorem , row2 column 1 , 2 x , column 2 equals , 1 half , open x plus 12 close , column 3 cap substitute . . , row3 column 1 , 4 x , column 2 equals x plus 12 , column 3 cap simplify , . , row4 column 1 , 3 x , column 2 equals 12 , column 3 cap subtract . x . fromeachside . . , row5 column 1 , x , column 2 equals 4 , column 3 cap divideeachsideby . 3. , end table

Exercises

Algebra Find the value of x.

4. 
5. 
6. Δ A B C cap delta eh b c  has vertices A ( 0 , 0 ) , B ( 2 , 2 ) , eh open 0 comma 0 close comma b open 2 comma 2 close comma  and C ( 5 , − 1 ) . c open 5 comma negative 1 close .  Find the coordinates of L, the midpoint of A C ¯ , eh c bar , comma  and M, the midpoint of B C ¯ . b c bar , .  Verify that L M ¯ ║ A B ¯ l m bar , box drawings double vertical , eh b bar  and L M = 1 2 A B . l m equals . 1 half eh b .

5-2 Perpendicular and Angle Bisectors

Quick Review

The Perpendicular Bisector Theorem together with its converse states that P is equidistant from A and B if and only if P is on the perpendicular bisector of A B ¯ . eh b bar , .

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

The Angle Bisector Theorem together with its converse states that P is equidistant from the sides of an angle if and only if P is on the angle bisector.

Example

In the figure, Q P = 4 q p equals 4  and A B = 8 . eh b equals 8 , .  Find QR and CB.

Q is on the bisector of ∠ A B C , angle eh b c comma  so Q R = Q P = 4 . q r equals . q p equals 4 , .

B is on the perpendicular bisector of A C ¯ , eh c bar , comma  so C B = A B = 8 . c b equals . eh b equals 8 , .

Exercises

7. Writing Describe how to find all the points on a baseball field that are equidistant from second base and third base.

In the figure, m ∠ D B E = 50 . bold italic m angle d b e equals 50 .  Find each of the following.

8. m ∠ B E D m angle b e d
9. m ∠ B E A m angle b e eh
10. x
11. y
12. BE
13. BC

Table of Contents