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?
To find the ratio CM/MB, we can use the Angle Bisector Theorem.
The Angle Bisector Theorem states that in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the adjacent sides.
In this case, we know that CN/NB = 21. According to the Angle Bisector Theorem, we have:
AN/AB = CN/AC
Since AN is an angle bisector of ∠BAC, we can assume that AN is divided into two segments such that AN = x and NB = y. Therefore, AB = x + y based on the properties of a triangle.
In our case, we are given that CN/NB = 21. Let's denote CN as 21z and NB as z.
So, we can rewrite the equation using the given values:
AN/AB = CN/AC
x / (x + y) = (21z) / (21z + z)
Simplifying the equation, we get:
x / (x + y) = 21z / 22z
Cross-multiplying, we have:
22zx = 21zx + 21zy
Subtracting 21zx from both sides:
zx = 21zy
Dividing by z:
x = 21y
Now, we can find the ratio CM/MB. We know that CM is perpendicular to AB since AM is the altitude. Therefore, CM + MB = y, based on the properties of a triangle.
Now, we substitute the value of x = 21y into the equation:
AM / AB = CM / MB
xA / AB = CM / (y - CM)
20y / 21y = CM / (y - CM)
Simplifying, we get:
20 / 21 = CM / (y - CM)
Cross-multiplying, we obtain:
20(y - CM) = 21CM
Expanding the left side:
20y - 20CM = 21CM
Adding 20CM to both sides:
20y = 41CM
Dividing by 41:
CM = 20y / 41
And since CM + MB = y, we substitute the value of CM:
20y / 41 + MB = y
Subtracting MB from both sides:
20y / 41 = y - MB
Then, isolating MB:
20y = 41y - 41MB
Adding 41MB to both sides:
41MB = 41y - 20y
Simplifying:
41MB = 21y
Dividing by 41:
MB = 21y / 41
Finally, we can find the ratio CM/MB:
CM/MB = (20y / 41) / (21y / 41)
Dividing by 41:
CM/MB = 20y / 21y
The y's cancel out:
CM/MB = 20 / 21
Therefore, the ratio CM/MB is 20/21.