A composite geometrical shape is made up of a square, equilateral and right triangles. Calculate the area of hatched triangle?

Question

Aurora Ayala · Accepted Answer

#156.25#

As #triangleABD# is a right triangle, we have, using the Pythagorean theorem, #bar(BD)=sqrt(25^2+25^2)=25sqrt(2)#. Because #triangleBDE# is equilateral, this implies that #bar(ED)=25sqrt(2)#.

Next, we can note that #triangleABD# is a #45-45-90# right triangle, meaning the angle #angleADB = 45^@#. Additionally, as #triangleBDE# is equilateral, we have #angleBDE=60^@#. Using those facts, along with #angleADF=180^@#, we have:

#angleEDF = angleADF - angleBDE-angleABD#

#=180^@-60^@-45^@#

#=75^@#

As #triangleEDF# is a right triangle, we can use the sine and cosine functions to say

#bar(EF)/bar(ED)=sin(75^@)#

#=>bar(EF)=bar(ED)sin(75^@)=25sqrt(2)sin(75^@)#

and

#bar(DF)/bar(ED)=cos(75^@)#

#=>bar(DF)=bar(ED)cos(75^@)=25sqrt(2)cos(75^@)#

As #bar(DF)# and #bar(EF)# are the base and height, respectively, of the hatched triangle, we can use the formula #A=1/2bh# to obtain the area of the hatched triangle as

#A =1/2bar(DF)*bar(EF)#

#=1/2(25sqrt(2)cos(75^@))(25sqrt(2)sin(75^@))#

#=625cos(75^@)sin(75^@)#

#=156.25#

Savannah Aguilar · Answer

undefined To calculate the area of the hatched triangle in a composite geometrical shape made up of a square, equilateral triangle, and right triangles, more information is needed. Specifically, the dimensions and configuration of the composite shape need to be provided to accurately determine the area of the hatched triangle. Without this information, it is not possible to calculate the area of the hatched triangle.