In an isosceles triangle with legs that are 1 unit long, the angles are 45 degrees, 67.5 degrees and 67.5 degrees What is its area?

| May 7, 2018

approximately ##0.35## square units
To find the area, we first need to find the height of the triangle, since the formula for area of a triangle is :
##Area_”triangle”=(base*height)/2##
First, we divide the isosceles triangle into ##2## right triangles.
Since we know that all right triangles have one ##90^@## angle and that all triangles have a ##180^@## sum of interior angles, then ##/_CAD## must be:
##/_CAD=180^@-90^@-67.5^@##
##/_CAD=22.5^@##
Using the Law of Sines, we can calculate the height of the right triangle:
##a/sinA=b/sinB=c/sinC##
##1/(sin90^@)=b/sin67.5^@##
##b*sin90^@=1*sin67.5##
##b*1=0.92##
##b=0.92##
Since we do not yet know the base length of the right triangle, we can also use the Law of Sines to find the base:
##a/sinA=b/sinB=c/sinC##
##1/(sin90^@)=c/sin22.5^@##
##c*sin90^@=1*sin22.5##
##c*1=0.38##
##c=0.38##
To find the base of the whole triangle, multiply the right triangle’s base length by ##2##:
##c=0.38*2##
##c=0.76##
Now that we have the base length and the height of the whole triangle, we can substitute these values into the formula for area of a triangle:
##Area_”triangle”=(base*height)/2##
##Area_”triangle”=((0.76)*(0.92))/2##
##Area_”triangle”=0.7/2##
##Area_”triangle”~~0.35##
##:.##, the area of the triangle is approximately ##0.35## square units.

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