Question 64167: In right triangle ACB (C is the right angle) CD is an altitude (D is between A and B). If AD=x^2 and BD=y^2, use Geometric Mean Theorems to find AC, BC, and CD interms of x and y (assume that x and y are positive)
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! In right triangle ACB (C is the right angle) CD is an altitude (D is between A and B). If AD=x^2 and BD=y^2, use Geometric Mean Theorems to find AC, BC, and CD interms of x and y (assume that x and y are positive)
FROM SIMILAR TRIANGLES ACB,ADC,CDB,WE GET
AC/AD=AB/AC....OR......AC^2 = AD*AB....................1
CB/DB=AB/CB....OR......CB^2 = AB*DB.......................2
DC/DB=AD/CD....OR.......CD^2 = AD.DB.....................3
FROM EQN.3
CD^2=X^2*Y^2
CD=XY...............4
FROM EQN.1
AC^2=X^2(AD+DB)=X^2(X^2+Y^2)
AC = X*SQRT(X^2+Y^2)
FROM EQN.2
CB^2=(AD+DB)*Y^2=(X^2+Y^2)Y^2
CB = Y*SQRT(X^2+Y^2)
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