The right triangle ABC, CD is the altitude to hypotenuse AB. If AC = 9, and AB = 27, Find AD

Question

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Answer 1

The altitude to the hypotenuse form three similar triangles.

Use that fact to determine ratios between the sides.

Answer 2

draw a diagram

label CD as y and label DB as x

first find CB

9^2+CB^2=27^2

81+CB^2=729

CB^2=729-81

CB^2=648

leave CB^2 as 648 to be used later

using triangle ADC...

9^2=(27-x)^2+y^2

81=729-54x+x^2+y^2

0=729-81-54x+x^2+y^2

0=648-54x+x^2+y^2

using triangle CDB...

x^2+y^2=CB^2

x^2+y^2=648 from above

x^2+y^2-648=0

x^2+y^2-54x+648=0

x^2+y^2-648=0

subtract the last two equations

-54x+1296=0

54x=1296

x=1296/54

x=24

DB=24 and therefore because AD=AB-DB, 27-24=3

AD=3  (answer)

also, CD=8.485 and CB=25.456