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.