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In triangle ABC, AB=AC and angle A  is equal to 36. Point D is on AC so that BD bisects angle ABC. 

(a) Prove that BC = BD = AD.

(b) Let x=BC and let y=CD. Using similar triangles BCD and ABC, write an equation involving x and y.

(c) Let r =y/x . Write the equation from Part (b) in terms of r, and findr 

(d) Find cos36 and cos72 using Parts a-c. (Do not use your calculator!)

 Jun 7, 2023
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(a) Since AB = AC and angle A = 36, we know that triangle ABC is an isosceles triangle. This means that angle B = angle C = 72. Since BD bisects angle ABC, we know that angle ABD = angle BDC = 18.

Now, let's look at triangle ABD. We know that angle ABD = 18 and angle A = 36. This means that angle ADB = 180 - 18 - 36 = 126.

Similarly, let's look at triangle BDC. We know that angle BDC = 18 and angle C = 36. This means that angle DCB = 180 - 18 - 36 = 126.

Since triangle ABD and triangle BDC are both isosceles triangles with the same base angles, we know that they are congruent. This means that AB = BD and AD = BD.

(b) Let x = BC and let y = CD. Using similar triangles BCD and ABC, we can write the following equation:

x/y = (AB + BD)/(BD) = AB/BD + 1

(c) Let r = y/x. Writing the equation from Part (b) in terms of r, we get:

r = AB/BD + 1

To find r, we need to find AB and BD. We know that AB = AC = x. We also know that BD = AD = y. Therefore, we can substitute these values into the equation for r to get:

r = x/y + 1 = 1 + 1 = 2

(d) To find cos36 and cos72, we can use the following identity:

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cos(A + B) = cosAcosB - sinAsinB

We can use this identity to find cos36 and cos72 as follows:

cos36 = cos(18 + 18) = cos18cos18 - sin18sin18

cos72 = cos(18 + 54) = cos18cos54 - sin18sin54

Since triangle ABC is an isosceles triangle, we know that angle B = angle C = 72. This means that angle A = 180 - 72 - 72 = 36.

Therefore, cos36 = cos(36) = cos18cos18 - sin18sin18.

Since r = 2, we know that sin18 = r/2 = 1. This means that sin18sin18 = 1/4.

We also know that cos18 = √(1 - sin^2(18)) = √(1 - 1/4) = √3/2.

Therefore, cos36 = cos18cos18 - sin18sin18 = (√3/2) * (√3/2) - (1/4) = 3/4 - 1/4 = 1/2.

Similarly, cos72 = cos(18 + 54) = cos18cos54 - sin18sin54.

Since r = 2, we know that sin54 = r/2 = 1. This means that sin54sin54 = 1/4.

We also know that cos54 = √(1 - sin^2(54)) = √(1 - 1/4) = √3/2.

Therefore, cos72 = cos18cos54 - sin18sin54 = (√3/2) * (√3/2) - (1/4) = 3/4 - 1/4 = 1/2.

 Jun 7, 2023

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