All of the following names of the polygon above are correct except for:

a.3 inches
b.12.16 inches
c.9 inches
d.9.16 inches

  B.  12.16 inches

Step-by-step explanation:

Triangle MNP has two sides given as length 5 (MN) and 10 (PN). The third side (MP) must be longer than 10* and less than 10+5=15. There is only one answer choice in that range: 12.16.

__

If we call the point of intersection of the diagonals X, then we know that NX = 8/2 = 4, and we can use the Pythagorean theorem to find MX and PX.

  MX² +NX² = MN²

  MX² + 4² = 5²

  MX = √(25 -16) = 3

__

  PX² +NX² = PN²

  PX² + 4² = 10²

  PX = √(100 -16)  = 2√21 ≈ 9.1652

Then ...

  MP = MX +PX = 3 +9.1652 = 12.1652 ≈ 12.17 . . . . . matches selection B

* The argument for the upper limit of MP comes from the triangle inequality. The argument for the lower limit comes from the observation that angle NMP is smaller than angle MNP, so side PN must be shorter than side MP. That is, MP must be greater than 10 inches.

or

step-by-step explanation:

we need to solve for 'x' the following equation:  

we have that:

⇒

⇒ ⇒

factorizing, we get that:

  ⇒

then, the solutions are: or

nothing extraneous has been found!