Altitude to the Hypotenuse Theorem - Tomorrow we will use this theorem to prove the Pythagorean Theorem!

Altitude to the Hypotenuse Theorem- Tomorrow we will use this theorem to prove the Pythagorean Theorem!

Altitude to Hypotenuse Theorem:

--the alt to hypotenuse forms two smaller right triangles that will be

similar to the original

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

--let’s color the smallest triangle, blue

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

-- next color the middle triangle red

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

--now let’s move & rotate the two small triangles to study all three at

the same time in the same orientation

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem:

Altitude to Hypotenuse Theorem

Altitude to Hypotenuse Theorem

Altitude to Hypotenuse Theorem

Altitude to Hypotenuse Theorem

Altitude to Hypotenuse Theorem

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

?

?

?

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

x

h

a

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

x

h

a

?

?

?

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

x

h

a

y

h

b

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

x

h

ac aa x

Altitude to Hypotenuse Theorem

x

y

c

h

b

a

y

h

bc byb

Altitude to Hypotenuse Theorem: either leg of the large triangle is the geom mean of

x

yh

b

a

c byb

c aa x

• the entire hypotenuse and

• the segment of the hyp adjacent to that leg.

Altitude to Hypotenuse Theorem

x

h

a

y

h

bx hyh

Altitude to Hypotenuse Theorem:

x

y

c

h

b

a

x

h

a

y

h

bx hyh

Altitude to Hypotenuse Theorem:--the alt to the hypotenuse is the geometric mean of the two segments of the hypotenuse.x

y

c

h

b

a

x hyh

Altitude to Hypotenuse Theorem:1. Alt to hyp forms 3 ~ rt triangles2. either leg of the large triangle is the

geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and

x

yc

h

b

a x hyh

3. the alt to the hyp is the geom mean of the two segments of the hypotenuse.

c byb

c aa x

Altitude to Hypotenuse Theorem:Sample Problem 1: (use part 2 of theorem)2. either leg of the large triangle is the

geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and

x10

6

c byb

c aa x

Altitude to Hypotenuse Theorem:Sample Problem 1: (use part 2 of theorem)2. either leg of the large triangle is the

geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and

x10

6

c byb

c aa x

10 66 x

Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of

the two segments of the hypotenuse.

4

yc

6

x hyh

Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of

the two segments of the hypotenuse.

4

yc

6

x hyh

4 66 y

Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of the two segments of the hypotenuse.

4

yc

6

x hyh

4 36

9

4 9 13

4 66

y

y

c

y

Altitude to Hypotenuse Theorem:Sample Problem 3: Find c and h.

x

6c

h12

Altitude to Hypotenuse Theorem:Sample Problem 3: Find h and c.

x

6

c

h12

6 144

24

24 6 18

1212 6c

c

x

c

Altitude to Hypotenuse Theorem:Sample Problem 3: Find h and c.

x

6

c

h12

6 144

24

24 6 18

1212 6c

c

x

c

2 108

108 6 3

186

h

h

hh