Final answer:
The tangent of 2X in triangle AXYZ is found using the double angle formula for tangent. After substituting the given values for the lengths of the sides (opposite = 4, adjacent = 3, hypotenuse = 5), we find the ratio that represents the tangent of 2X to be -8/7.
Step-by-step explanation:
To find the tangent of 2X for triangle AXYZ, where the measure of angle Z is 90°, ZY (which could be side b or the opposite side for angle X) equals 4, XZ (which could be side a or the adjacent side for angle X) equals 3, and YX (which would be the hypotenuse c) equals 5, we can use the tangent double angle formula.
The tangent of angle X, given a right triangle, is the length of the opposite side divided by the length of the adjacent side, expressed as tan(X) = opposite/adjacent. In the context of triangle AXYZ, this would be tan(X) = ZY/XZ = 4/3.
The double angle formula for tangent is expressed as:
tan(2X) = 2 an(X) / (1 - tan2(X))
Plugging in the values we have:
tan(2X) = 2*(4/3) / (1 - (4/3)2)
Now, simplify the expression:
tan(2X) = 8/3 / (1 - 16/9)
tan(2X) = 8/3 / (-7/9)
tan(2X) = -24/21
tan(2X) = -8/7
Hence, the ratio that represents the tangent of 2X in triangle AXYZ is -8/7.