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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?

    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.