In the triangle embedded in the square what is the measure of angle, #theta#?

Answer 1

#theta = arctan((3A-sqrt(A))/(3A+sqrt(A)))#

Let #s# represent the side length of the square.
From the right side, we see that #s = sqrt(A)+bar(ED)#
From the area of the triangle, we have #A=(s*bar(ED))/2#. Solving for #bar(ED)# gives us #bar(ED) = (2A)/s#. Substituting this into the above leaves us with #s = sqrt(A)+(2A)/s#.
With our new equation in #s# and #A#, we can multiply both sides by #s# and gather the terms on one side to obtain the quadratic
#s^2-sqrt(A)s-2A = 0#

Applying the quadratic formula gives us

#s = (sqrt(A)+-3A)/2#
As we know #s > sqrt(A)# we can discard #(sqrt(A)-3A)/2#, leaving us with
#s = (sqrt(A)+3A)/2#
We can substitute this value for #s# into the equation obtained from the right side of the square to obtain
#(sqrt(A)+3A)/2 = sqrt(A)+bar(ED)#
#=> bar(ED) = (3A-sqrt(A))/2#
Now, as #triangleAED# is a right triangle, we have
#tan(theta)=bar(ED)/bar(AD)#
#=bar(ED)/s#
#=(3A-sqrt(A))/2-:(sqrt(A)+3A)/2#
#=(3A-sqrt(A))/(3A+sqrt(A))#

Thus, taking the inverse tangent function of both sides, we get the result

#theta = arctan((3A-sqrt(A))/(3A+sqrt(A)))#
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Answer 2

Without specific information about the triangle or square, such as the lengths of their sides or any angles given, it's not possible to determine the measure of angle theta. The measure of angle theta depends on the specific properties and dimensions of the triangle and square involved.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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