SOLUTION: Triangle ACB is in inscribed in cicle o. Given: AB is diameter and CD is perpendicular to AB. Prove: (CD)^2=(AD)(DB).

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Question 193358This question is from textbook Essentials of Geometry for College Students
: Triangle ACB is in inscribed in cicle o. Given: AB is diameter and CD is perpendicular to AB. Prove: (CD)^2=(AD)(DB).This question is from textbook Essentials of Geometry for College Students

Answer by jojo14344(1512) About Me  (Show Source):
You can put this solution on YOUR website!


drawing%28350%2C350%2C-4%2C4%2C-4%2C4%2Ccircle%280%2C0%2C3%29%2Ctriangle%28-3%2C0%2C0%2C3%2C3%2C0%29%2Clocate%28-3.2%2C0%2CA%29%2Clocate%280%2C3.3%2CC%29%2Clocate%283.2%2C0%2CB%29%2Cgreen%28line%280%2C0%2C0%2C3%29%29%2Clocate%280%2C-.3%2CD%29%2Cgreen%28line%28-.2%2C1.5%2C.2%2C1.5%29%29%2Cgreen%28line%28-1.5%2C.2%2C-1.5%2C-.2%29%29%2Cgreen%28line%281.5%2C.2%2C1.5%2C-.2%29%29%29
Prove: CD%5E2=%28AD%29%28AB%29

We know Diameter=AB, and Radius=%281%2F2%29Diameter=%281%2F2%29AB

We can see the triangle is split into 2 Isosceles Right Triangles.
Theorem:
In an isosceles right triangle, the sides are in the ratio 1%3A1%3Asqrt%282%29
The equal sides make the right angle.

Therefore, equal sides ---> AD=DB=CD=Radius=(1/2)(AB)

Proving: %28CD%29%5E2=%28AD%29%28DB%29
=%28AD%29%5E2=%28AD%29%28AD%29 --> %28AD%29%5E2=%28CD%29%28DB%29
=%28DB%29%5E2=%28DB%29%28DB%29 --> %28DB%29%5E2=%28CD%29%28AD%29
=%28CD%29%5E2=%28CD%29%28CD%29 --> highlight%28%28CD%29%5E2=%28AD%29%28DB%29%29; *Note: AD=DB=CD=Radius=(1/2)(AB)

Thank you,
Jojo