Question

Let ABC be a right triangle with a right angle at C. Let D and E be points on AB with D between A and E such that CD and CE trisect C. If DE = 8 and BE = 15, then tan B can be written as where m and n are relatively prime positive integers, and p is a positive integer not divisible by the square of any prime. Find m + n + p.

Answer 1

To find tan(B), use the tangent ratio with the given lengths of DE and CD.

Step-by-step explanation:

To find tan(B), we can use the relevant trigonometric ratios in a right triangle. Let's start by drawing the triangle ABC, with a right angle at C. We know that CD and CE trisect angle C, so angle CDE and angle CED are each 30 degrees.

Since DE = 8 and CE = 15, we can use the law of sines to find the length of CD: 15 / sin(30) = CD / sin(120).

Solving for CD, we find that CD = 15 * sin(120) / sin(30) = 25.

Now that we know the lengths of all three sides of triangle CDE, we can use the tangent ratio: tan(B) = DE / CD = 8 / 25.