In a right triangle ABC, CD is an altitude, such that AD=BC. Find AC, if AB=3 cm, and CD= 2 cm.

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Question:

In a right triangle ABC, CD is an altitude, such that AD=BC. Find AC, if AB=3 cm, and CD= 2 cm.

Answer:

Consider right triangle ΔABC with legs AC and BC and hypotenuse AB. Draw the altitude CD.

1. Theorem: The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.

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According to this theorem,

?f=BC%5E2%3DBD%5Ccdot%20AB

Let BC=x cm, then AD=BC=x cm and BD=AB-AD=3-x cm. Then

?f=x%5E2%3D(3 x)%5Ccdot%203%2C%5C%5C%20%5C%5Cx%5E2%3D9 3x%2C%5C%5C%20%5C%5Cx%5E2%2B3x 9%3D0%2C%5C%5C%20%5C%5CD%3D3%5E2 4%5Ccdot%20( 9)%3D9%2B36%3D45%2C%5C%5C%20%5C%5C%5Csqrt%7BD%7D%3D%5Csqrt%7B45%7D%3D3%5Csqrt%7B5%7D%2C%5C%5C%20%5C%5Cx 1%3D%5Cdfrac%7B 3 3%5Csqrt%7B5%7D%20%7D%7B2%7D%3C0%2C%5C%20x 2%3D%5Cdfrac%7B 3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%3E0

Take positive value x. You get

?f=AD%3DBC%3D%5Cdfrac%7B 3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%5C%20cm

2. According to the previous theorem,

?f=AC%5E2%3DAD%5Ccdot%20AB

Then

?f=AC%5E2%3D%5Cdfrac%7B 3%2B3%5Csqrt%7B5%7D%20%7D%7B2%7D%5Ccdot%203%3D%5Cdfrac%7B 9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%2C%5C%5C%20%5C%5CAC%3D%5Csqrt%7B%5Cdfrac%7B 9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%7D%5C%20cm

Answer: ?f=AC%3D%5Csqrt%7B%5Cdfrac%7B 9%2B9%5Csqrt%7B5%7D%20%7D%7B2%7D%7D%5C%20cm

This solution doesn’t need CD=2 cm. Note that if AB=3cm and CD=2cm, then

?f=CD%5E2%3DAD%5Ccdot%20DB%2C%5C%5C%20%5C%5C2%5E2%3DAD%5Ccdot%20(3 AD)%2C%5C%5C%20%5C%5CAD%5E2 3AD%2B4%3D0%2C%5C%5C%20%5C%5CD%3C0

This means that you cannot find solutions of this equation. Then CD≠2 cm.

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