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| A 45 degree angle within a triangle (Posted on 2024-11-25) |
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Let ABC be a right triangle with its right angle at C and side AC=3. Let point D be a point on side BC such that BD=5 and angle BAD=45 degrees.
How long is CD?
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Submitted by Brian Smith
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Solution:
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Construct point E on side AB such that angle ADE=90degrees. Then construct point F such that EF is perpendicular to BC.
Then AD=DE, angle CAD = angle FDE, and angle ACD = angle DFE. Therefore triangles ACD and DFE are congruent. Then DF=3 and BF=2.
Angle ACB = angle EFB and angle FBE is the same as angle CBA. Therefore triangles ABC and EBF are similar. Let CD=x. Then 3/(x+5) = x/2.
Simplifying the equation yields x^2+5x-6=0. Then (x-1)*(x+6)=0, or x=-6 or x=1. x=-6 is rejected so that leaves x=1 as the length of CD. |
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