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Explanation:</h3>
All of the right triangles are similar by AA similarity, so corresponding side lengths are proportional. The ratio of the long leg to the short leg is the same for the two smaller triangles, for example:
CP/AP = MP/CP
CP/9 = 16/CP . . . . . fill in the given numbers
CP² = 9·16 . . . . . . . multiply by 9·CP
CP = 3·4 = 12 . . . . . take the square root
Now, you can use the Pythagorean theorem to find AC and/or CM.
AC = √(9² +12²) = √225 = 15
CM = √(12² +16²) = √400 = 20
In summary, CP = 12, AC = 15, CM = 20.
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Once you have CM, you can see these are 3-4-5 right triangles, so you can determine the other lengths by using these side ratios.
3:4:5 = 9:12:15 = 12:16:20
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The altitude CP is called the "geometric mean" of AP and MP. It is the square root of their product. This is true for <em>any</em> right triangle, not just one with sides in the ratio 3:4:5. If you know this, you can write down your answers almost immediately. Above, we had to derive this fact using similarity.
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