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Section 4-7

- Medians, Altitudes and Perpendicular Bisectors

Median

- connects the vertex to the midpoint of the

opposite side

Thus, every triangle has three medians.

Altitude

- the perpendicular segment from a vertex to a line

that contains the opposite side

B

F

D

C

A

E

- For obtuse triangles, two altitudes fall outside

the figure. - You extend the base of the triangle so that it

intersects the altitude at a right angle.

B

B

C

A

C

H

A

J

For obtuse triangles, there is still one altitude

in the triangle.

B

K

C

A

In Right Triangles Two of the altitudes lie on

the legs of the triangle. The 3rd is inside.

Perpendicular bisector of a

segment

- is a line (or ray or segment) that is

perpendicular to the bisector at its midpoint

In a given plane, there is exactly one

perpendicular to a segment at its midpoint

M

l

K

J

Theorem 4-5

- If a point lies on the perpendicular bisector of

a segment, then the point is equidistant from the

endpoints of the segment

Theorem 4-6

- If a point is equidistant from the endpoints of a

segment, then the point lies on the perpendicular

bisector of the segment.

A

B

C

X

Theorem 4-7

- If a point lies on the bisector of an angle,

then the point is equidistant from the sides of

the angle.

Theorem 4-8

If a point is equidistant from the sides of an

angle, then the point lies on the bisector of the

angle.

Always, Sometimes or Never?

Always

- An altitude is _______ perpendicular to the

opposite side? - A median is _______ perpendicular to the

opposite side? - An altitude is ________an angle bisector?

Sometimes

Sometimes

Always, Sometimes or Never?

Sometimes

- An angle bisector is ___________ perpendicular

to the opposite side. - A perpendicular bisector is ________

perpendicular to a segment at its midpoint. - A perpendicular bisector of a segment is

__________ equidistant from the endpoints of the

segment.

Always

Always

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