<p>"P796, #20."</p>
<p>You do not need trig. First, all 5 lengths are equal:give them a value then. I chose 1, but you can chose "x" or "z" or "50". </p>
<p>1)So triangles ABD and DBC are congruent, first of all (SSS.) Same with triangles ABC and ADC (SSS.)<br>
2) Since triangle BDC's sides are all equal, that means all angles are equal. Each angle in triangle BDC is equal to 60.
3)Since BDC and ABD are congruent triangles, angle DBC and angle ABD are equal; angle DBC + a ABD = 120. So angle ABC is =120. When you look @ triangle ABC, you will notice that it is isosceles. So if ABC is 120, angle BAC and angle ACB will be equal and therefore each will be 30 degrees.
4)So, the question asks the ration between the length of AC (which we will find next) and BD (we have given this the value "1".)
5) We already know that angle ABD is equal to 60.
6) Since line BD bisects angle ABC and CDA, we know that it is perpendicular to AC. So angle AEC (E where the two diagonals intersect) is 90. We then know that triangle AEB is a 30 60 90 triangle.
7) Thats all you need to know; AB is equal to 1, BD must be equal to 1/2, AE must be equal to 1/2rad3. If AE is 1/2rad3, then AC must be 2 times that, or just rad 3. we know that BD is one. So there we go, the ratio is rad3 to 1.</p>