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 �ÚBAC. If CN/NB =21 , what is CM/MB ?
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To find the ratio CM/MB, we need to determine the lengths of CM and MB first.
Let's start by labeling the lengths of CN and NB as x and y, respectively. We are given that CN/NB = 21, which means CN = 21 * NB.
Since AN is the angle bisector of angle BAC, we can use the angle bisector theorem to find the ratio of lengths BM/MA. According to the angle bisector theorem, BM/MA = BN/NA.
Since AN is the altitude of triangle ABC, we know that BN = BA + AN. Similarly, MA = BA - AN.
Given that triangle ABC is a right-angled triangle at A, we can use the Pythagorean theorem to find BA.
Let's assume that AB = c and AC = b. By the Pythagorean theorem, we have:
c^2 = b^2 + AB^2
Since triangle ABC is right-angled at A, we have:
c^2 = b^2 + c^2
Simplifying the equation, we find:
b^2 = 0
Thus, b = 0, which means AB = c.
Since AB = c and AN is the angle bisector, we can write:
BN = c + AN
MA = c - AN
Substituting these expressions into BM/MA = BN/NA, we get:
BM/(c - AN) = BN/AN
Cross-multiplying, we find:
BM * AN = BN * (c - AN)
Expanding the right side of the equation, we have:
BM * AN = (c + AN) * (c - AN)
Simplifying further, we get:
BM * AN = c^2 - AN^2
Since AB = c, we can rewrite c^2 as AB^2:
BM * AN = AB^2 - AN^2
Using the Pythagorean theorem for triangle AMN, we know that:
AN^2 + MN^2 = AM^2
Since AM is the altitude, MN = BM. Substituting this relationship into the equation, we have:
AN^2 + BM^2 = AM^2
Rearranging, we find:
BM^2 = AM^2 - AN^2
Substituting this expression for BM^2 into the earlier equation, we have:
BM * AN = AB^2 - BM^2
Rearranging further, we get:
BM^2 + BM * AN - AB^2 = 0
This is a quadratic equation in terms of BM. We can solve it using the quadratic formula:
BM = (-AN + √(AN^2 + 4 * AB^2)) / 2
Now, substituting the values we know:
BM = (-AN + √(AN^2 + 4 * c^2)) / 2
Given that CN = 21 * NB, we can add these lengths:
BC = CN + NB
BC = 21 * NB + NB
BC = 22 * NB
Since BM = CN and CM = BC - BN, we can substitute these values:
CM = BC - BN
CM = 22 * NB - BM
Now, let's solve for CM/MB:
CM/MB = (22 * NB - BM) / (-AN + √(AN^2 + 4 * c^2)) / 2
In order to find the exact value of CM/MB, we need the values of AN, c, and NB.