5-3 Medians and Altitudes - 5-3 CONCURRENT LINES, MEDIANS, AND

5-3 CONCURRENT LINES, MEDIANS, AND ALTITUDES (p. 256-263)



Note: Just cover the definitions of median and altitude in this section. Do some

algebra problems relating to these segments.



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

opposite side.



Example: Is AD a median in ABC?



BD = 2.66 cm A

DC = 2.66 cm









B D C





How many medians does a triangle have?



Example: In the following triangle, AD is a median, DC  7x - 13, and BC  5x  25.

Set up and solve an equation to find x. Write x as a fraction. Then, find BC and BD.

A









B D C





Example: In the following diagram, D, E, and F are midpoints of the three sides of the

triangle. What are the segments AD, BE, and CF called? Point X is called the centroid

of ABC. Do you think that the centroid is the center of gravity of this triangle?



A





E

F

X



B D C

An altitude of a triangle is a segment that starts at a vertex and is perpendicular to the

opposite side. For certain triangles, however, the opposite side must be extended or

lengthened in order to obtain perpendicularity.



Example: In the following diagram, what kind of special segment is AE ? Why? What

kind of special segment is AD ? Why?



BE = 2.61 cm A

EC = 2.61 cm

mAEC = 64.70

BD = 4.21 cm

mADC = 90.00

DC = 1.01 cm







B E D C





Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or

it may even lie outside of a triangle.



How many altitudes does a triangle have?



For the following acute triangle, can you name the three altitudes? Where do the three

altitudes lie?

mCYB = 90.00

mBXA = 90.00 C

mAZB = 90.00

X Z

G









A

Y

B





Complete the sentence. For an acute triangle, the three altitudes lie __________ the

triangle.

For the following obtuse triangle, can you name the three altitudes? Where do the three

altitudes lie?





mBAC = 112.70 

K mADC = 90.00

H mBHA = 90.00

mCKA = 90.00

A









C

D

B









Complete the sentence. For an obtuse triangle, one altitude lies __________ the triangle

and two altitudes lie __________ the triangle.



For the following right triangle, can you name the three altitudes? Where do the three

altitudes lie?





mADB = 90.00

C

D mCAB = 90.00

mADB = 90.00





A B

Complete the sentence. For a right triangle, one altitude lies __________ the triangle and

two altitudes lie ____________ the triangle.

Example: In the following diagram, m CZA  4x - 25 and CB  2x  15. Set up and

solve an equation to find x. Then, find CB.

mCYB = 90.00

mBXA = 90.00 C

mAZB = 90.00

X Z

G









A

Y

B





Homework p. 259-263: 14-16,19-22,27-29,43,44,46,48,51