# Thread: Year 10 - Geometry/Trigonometry

1. ## Year 10 - Geometry/Trigonometry

http://i.stack.imgur.com/nD2Lxl.jpg

Please ignore the pencilled 4m in the diagram but I really need to know what the length of the bottom line - line DC - is. A procedure or tips on how to calculate this would be useful. Also, is the pencil line from point C dividing right angle ACD exactly in half? I figured this would be useful to calculate the solution to this problem but I doubt that it is dividing right angle ACD exactly in half. Please use correct rules and explain all working and reasons for this.
(Try not to use quadratic equations and such unless you explain exactly what this means as I have no idea what quadratic equations are and am only in year 10. I do however know basic trigonometric functions such as sin, cosin, tan, and how to use these to find angles. I also know Pythagoras theorem etc. obviously)

2. ## Re: Year 10 - Geometry/Trigonometry

Are you familioar with the identity for the tangent of added angles?

$\displaystyle \tan (a+b) = \frac {\tan a - tan b}{1-\tan a \tan b}$

Call the small angle x, and the unknown length CD = L. Then:

1/L = tan x
4/L = tan(x+35)

Hence: tan(x+35) = 4 tan x

Now use the formula for tan(x+35) to get:

$\displaystyle \frac {\tan x + \tan(35)}{1- \tan x \tan(35)}$

Now solve for $\displaystyle \tan x$. Almost done: You know that 1/L = tan x, so solve for L.

As for the line you drew from C that intersects with the hypotenuse - no, it is not a bisector unless the right triangle is isocoles. Consider that a bisector of a right angle is 45 degees, and the perpendicular intersection of the hypotenuse is 90 degrees. Since angles of a triangle add to 180, the angle A would have to be 45 degrees for this to work.