circumcenter, find the length AU

[hongw]

Let ABC be an isosceles triangle with AB=AC. Let AB=10 and BC=12. Let G be the centroid of this triangle, and let U be the circumcenter.

1a. Find the length of the altitude from A to BC. (answer 8)
1b. Find the area of the triangle. (answer 48)
1c. Find the length of the altitude from C to AB. (answer 9.6)

1d. Find the length AU. (?????????????? need help please!!!!!!!!!!!!!!!!)

 

[pappus]

Let ABC be an isosceles triangle with AB=AC. Let AB=10 and BC=12. Let G be the centroid of this triangle, and let U be the circumcenter.

1a. Find the length of the altitude from A to BC. (answer 8)
1b. Find the area of the triangle. (answer 48)
1c. Find the length of the altitude from C to AB. (answer 9.6)

1d. Find the length AU. (?????????????? need help please!!!!!!!!!!!!!!!!)View attachment 1724

1. Unfortunately I can't read your notes.

2. I assume that you have already calculated the value of angle at A(?): I've got \(\displaystyle \angle(A) \approx 73.73979°\)

3. The angle at the centre U in the triangle UCM is twice as large as A. Use the indicated right triangle to determine the length of r. I've got r = 6.25

EDIT: Much to my surprise I was able to see your notes.

The height \(\displaystyle h_A\) from A to BC is 8. (Why?) Use the indicated right triangle in the 2nd image (you should recognize it ). Use the Pythagorean theorem:

\(\displaystyle 6^2+(h_A-r)^2=r^2\) which yields the same value for r.

 

[hongw]

thank you very much, i have to digest it.