What is CD?

Question

In the ΔABC, AB = 16 in, BC = 9 in, AC = 10 in. AD ⊥ to the extended BC. Find CD.

 

A \^----------

    \           ^{------------------}

     \                            ^{---------------}

  C   \_________________________ B

 

 

that is supposed to be the shape of the triangle (if I am Correct)

Answer 1

draw the diagram and extend BC to D

draw AD

use the Pythagorean Theorem with triangle ADC

call AD "h"

call DC "x"

h²+x²=10²

now use the Pythagorean Theorem with triangle ADB

AD=h, AB=16, DB=(x+9)

h²+(x+9)²=16²

from the first equation get h²=10²-x²

from the second equation get h²=16²-(x+9)²

set the two right sides equal to each other because they both equal h²

10²-x²=16²-(x+9)²

100-x²=256-x²-18x-81

cancel -x² from both sides

100=256-18x-81

100=175-18x

100-175=-18x

-75=-18x

18x=75

x=75/18

x=4.1666... inches or 4 1/6 inches for the length of DC

 

 

Answer 2

Yes, for your question about ABC being a triangle.  If BC is extended to point D (directly under point A), you create a right triangle with a new base (BC + x).  Therefore, the base of the smaller triangle is (9 + x), height h and the hypotenuse of length 10.  How would you proceed?