perplexus.info :: Geometry : Equilateral and Isosceles

Equilateral and Isosceles (Posted on 2025-05-30)

Let triangle ABC and a point D on AC such that BD = DC = 3. If AD = 6 and ∠ ACB = 30◦, calculate ∠ ABD.

No Solution Yet Submitted by Danish Ahmed Khan

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

Solution

Comment 1 of 1

Triangle BDC is a 30-30-120 isosceles triangle, then BC=3*sqrt(3).

Let point E be the base of the altitude from B onto side AC.

BEC is a 30-60-90 right triangle. Then BE=(3/2)*sqrt(3)

BED is a 30-60-90 right triangle. Then EC=9/2.

Then DE=3/2 and AE=9/2

AEB is a right triangle with legs AE and EB. AE/EB=sqrt(3) so it is a 30-60-90 triangle.

Angle ABE=60 and angle EBD=30, so angle ABD=90 degrees.

Posted by Brian Smith on 2025-05-31 17:49:45

Please log in:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info