Find the value of each variable

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  11.  x = 13, y = 13√2

  12.  x = 8, y = 4√3

  13.  x = 14√3, y = 7√3

  14.  x = y = 11√2

  15.  x = 12√2, y = 12, z = 12√3

Step-by-step explanation:

The ratios of side lengths in the "special" triangles are (shortest to longest)

  1 : 1 : √2 . . . . . isosceles right triangle

  1 : √3 : 2 . . . . 30°-60°-90° triangle

These ratios make answering these questions much easier.

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11. 13 : x : y = 1 : 1 : √2

  x = 13, y = 13√2

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12. 4 : y : x = 1 : √3 : 2

  x = 8, y = 4√3

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13. y : 21 : x = 1 : √3 : 2

  y = 21/√3 = 7√3

  x = 14√3

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14. x : y : 22 = 1 : 1 : √2

  x = 22/√2 = 11√2

  x = y = 11√2

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15. Let 'a' represent the altitude. Then we have ...

  a : z : 24 = 1 : √3 : 2

  a = 12, z = 12√3

and ...

  a : y : x = 1 : 1 : √2

  y = 12, x = 12√2

Additional comment

It is worth remembering that 1/√n = (√n)/n. This is useful when you have to divide by √2 or √3 in the above problems.

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions

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