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: x y c h b a

x y c h b a ? ? ?

x y c h b a x h a

x y c h b a x h a ? ? ?

x y c h b a x h a y h b

x y c h b a x h a

Altitude to Hypotenuse Theorem x y c h b a y h b

Altitude to Hypotenuse Theorem: either leg of the large triangle is the geom mean of x y h b a the entire hypotenuse and the segment of the hyp adjacent to that leg.

Altitude to Hypotenuse Theorem x h a y h b

Altitude to Hypotenuse Theorem: x y c h b a x h a y h b

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

Altitude to Hypotenuse Theorem: 1. Alt to hyp forms 3 ~ rt triangles 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  x y c h b a 3.the alt to the hyp is the geom mean of the two segments of the hypotenuse. 

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  x 10 6

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  x 10 6

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 y c 6

4 y c 6

4 y c 6

Altitude to Hypotenuse Theorem: Sample Problem 3: Find c and h. x 6 c h 12

Altitude to Hypotenuse Theorem: Sample Problem 3: Find h and c. x 6 c h 12

Altitude to Hypotenuse Theorem: Sample Problem 3: Find h and c. x 6 c h 12