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!)
(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:
Use code with caution. Learn more
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.
