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. AA2. SAS3. SSS

ActivityTake 10 minutes.Use graph paperDraw a right triangleDraw the altitude from the right angleProve 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

projection

leg

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 leg2. Measurement of projection3. Measurement of hypotenuse

projection leg

leg hypotenuse

leg

proj

leg

proj

projection leg

leg hypotenuse

leg

proj

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. Altitude2. 2 segments that determine hypotenuse

i.e. projections

projection altitude

altitude projection

proj proj

altitude

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. Hypotenuse2. Altitude3. Length of legs

hypotenuse altitude leg leg

leg leg

hypotenuse

altitude

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.