ABC is a triangle with a right angle at
A.M and N are points on BC such that AM is the altitude, and AN is the
angle bisector of ∠BAC. If CN/NB=21,what is CM/MB?
Have been working on it, hard sketch to make, unless I am missing something.
so far I have:
BC = 22
AB = √242/29
AC = 41√242/29
sin (angle ANC) = exactly 22/29
(when I sketched this, it made no sense)
will not be able to get back to it until tonight, if you have the patience to wait.
I have a solution, but what a mess.
Before I type it all out, let me know if you still need the solution. I certainly don't feel like typing it all out unless you will actually look at it.
Yes please Reiny. It would help me a lot. Thanks! :)
To find CM/MB, we need to first understand the properties of the angle bisector and the altitude in a right-angled triangle.
In a triangle, the angle bisector divides the opposite side into segments that are proportional to the adjacent sides. This property is known as the Angle Bisector Theorem.
Therefore, we have AN/NB = AC/CB.
Similarly, the altitude of a right-angled triangle forms two similar right-angled triangles with the original triangle. This property allows us to establish the following relationship:
AM/MB = AC/CB.
Since AM is the altitude and AN is the angle bisector, we can combine these two relationships as follows:
AM/MB = AC/CB = AN/NB.
Given that CN/NB = 21, we can substitute this value into the equation:
AM/MB = AN/NB = AC/CB = AN/(21).
To find CM/MB, we need to express CM in terms of AM and MB. Since AM and CM make up the entire side AC, we have:
AM + CM = AC.
Rearranging this equation, we can express CM in terms of AM and MB:
CM = AC - AM = CB - MB.
Substituting this value into the equation, we get:
AM/MB = AN/(21) = (CB - MB)/(21).
Now, let's solve for CM/MB:
CM/MB = (CB - MB)/(21MB).
Simplifying the equation, we have:
CM/MB = CB/(21MB) - 1/21.
Since we know that CN/NB = 21, we can substitute CN = 21NB to get:
CM/MB = (CN + NB)/(21MB) - 1/21.
Now, substituting CN = AC - AN and NB = AC - AN into the equation, we have:
CM/MB = [(AC - AN) + (AC - AN)]/(21MB) - 1/21.
Since AC = AM + CM, we can substitute this value and simplify further:
CM/MB = [(AM + CM - AN) + (AM + CM - AN)]/(21MB) - 1/21.
Simplifying, we get:
CM/MB = (2AM - 2AN)/(21MB) - 1/21.
Finally, we can simplify this expression further by dividing both terms by 2:
CM/MB = AM/21MB - AN/21MB - 1/21.
Therefore, CM/MB can be expressed as:
CM/MB = (AM - AN)/(21MB) - 1/21.
This is the final expression for CM/MB in terms of AM, AN, and MB, given that CN/NB = 21.