Transcript Metric Relations in Right Triangles


By drawing the altitude
from the right angle of
a right triangle, three
similar right triangles
are formed
C
Corresponding angles
are congruent
AND
 Corresponding sides
are proportional in
length

Minimum Conditions:
1. AA
2. SAS
3. SSS

Activity
Take 10 minutes.
Use graph paper
Draw a right triangle
Draw the altitude from the right angle
Prove that 3 similar triangles are formed


Activity 1: Birds of a feather stick together!
Problem: Mother Nature Enraged!

The altitude to the
hypotenuse of a right
triangle forms two
triangles that are
similar to each other
and to the original
triangle.
leg
leg
projection
projection

Using the lengths of the corresponding sides
of the triangles formed, we can determine the
ratios and from this determine certain
geometric properties
Information we have or need:
1. Measurement of leg
2. Measurement of projection
3. Measurement of hypotenuse

leg
leg
proj
proj
projection
leg

leg
hypotenuse
leg
proj
projection
leg

leg
hypotenuse
leg
proj

In a right triangle the length of the leg of a
right triangle is the geometric mean between
the length of its projection on the hypotenuse

Worksheet hand out on Property 1
Information we have or need:
1. Altitude
2. 2 segments that determine hypotenuse i.e.
projections

altitude
proj
proj
projection
altitude

altitude
projection

In a right triangle the
length of the altitude
drawn from the right
angle is the geometric
mean of the length of
the two segments that
determine the
hypotenuse

Worksheet on Property 2
Information we have or need:
1. Hypotenuse
2. Altitude
3. Length of legs

leg
leg
altitude
hypotenuse
hypotenuse  altitude  leg  leg

In a right triangle, the
product of the length
of the hypotenuse and
its corresponding
altitude is equal to the
product of the lengths
of the legs.

Hand out on property 3

Pythagorean Theorem

Visions page 182, numbers 1,2,3,4

Each leg of a right triangle is the mean
proportional between the hypotenuse and the
projection of the leg on the hypotenuse.

or

The altitude to the hypotenuse of a right
triangle is the mean proportional between the
segments into which it divides the
hypotenuse.