SOLUTION: In ΔABC, m∠ACB = 90�, CD ⊥ AB and m∠ACD = 45�. Find: Find AC, if CD = 6(root)2 in.

Question 1065915: In ΔABC, m∠ACB = 90�,
CD ⊥ AB and m∠ACD = 45�. Find:
Find AC, if CD = 6(root)2 in.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
m∠ACB = 90� tells us that ABC is a right triangle.
CD ⊥ AB tells us that ACD and BCD are right triangles too.
In other words, m∠ADC = 90� and m∠BDC = 90�.
m∠ACD = 45�, together with m∠ADC = 90�,
tells us that m∠DAC = 90� - 45� = 45� .
So ADC is an isosceles right triangle,
with two congruent 45� base angles,
and two congruent legs, measuring .
In every isosceles right triangle,
the ratio of hypotenuse to leg length is ,
so I need not think much to know for triangle ADC the hypotenuse length is
.

Alternative ways to get to the solution:
Knowing that ADC is an isosceles right triangle,
we could apply the Pythagorean theorem
to calculate the length hypotenuse AC, in inches, as

Otherwise, knowing that ADC is a right triangle with
m∠ACD = 45� ,
we can calculate the trigonometric ratio for angle ACD
cos(ACD)=(adjacent side)/hypotenuse
and substituting known values,

Dividing vt both sides of the equal sign,
,
and multiplying both sides times

.

NOTE:
BY the way, all those triangles are isosceles right triangles,
DCB is congruent with ACD, and ABC is similar, to DCB and DCA,
larger by a scale factor od , of course.
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SOLUTION: In &#916;ABC, m&#8736;ACB = 90�, CD &#8869; AB and m&#8736;ACD = 45�. Find: Find AC, if CD = 6(root)2 in.

SOLUTION: In ΔABC, m∠ACB = 90�, CD ⊥ AB and m∠ACD = 45�. Find: Find AC, if CD = 6(root)2 in.