Answer:
x = 9
Explanation:
Given a figure showing similar right triangles, you want x as the segment of the hypotenuse touching the side of length 3√19, where the other hypotenuse segment is 10.
Proportion
The ratio of the short side to the hypotenuse is the same for similar triangles AMH and ATM.
MA/TA = HA/MA
3√19/x = (10+x)/(3√19)
9·19 = x(x+10)
We observe that the factors 9 and 19 differ by 10, which is also the difference between x and x+10. This tells us x=9.
(We could solve the quadratic by looking for factors of the product 9·19 = 171 that differ by 10. The positive solution would be x=9.)
The value of x is 9.
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Additional comment
We have written the problem statement using the words "touching the side of length 9√19" because this is key to the use of the "geometric mean relation" expressed by ...
√(x(x+10)) = 9√19
That is, the geometric mean of the segments (AT, AM) touching segment MA is equal to the length of MA. The geometric mean of two numbers is the square root of their product.