Answer:
Similar triangles means that they have proportional sides and congruent angles.
According to the similarity, we can form the following proportion between corresponing sides
[tex]\frac{7x}{5x+2}=\frac{XY}{21}[/tex]
However, we need to find an expression for XY using Pythagorean's Theorem:
[tex](7x)^{2} =28^{2} +XY^{2}\\XY^{2}=49x^{2} -784\\XY=\sqrt{49(x^{2}-16)} =7\sqrt{x^{2} -16}[/tex]
Replacing it in the proportion, we have
[tex]\frac{7x}{5x+2}=\frac{7\sqrt{x^{2}-16 } }{21}\\21x=(5x+2)\sqrt{x^{2}-16 }\\(21x)^{2} =(5x+2)^{2}(\sqrt{x^{2}-16 })^{2} \\441x^{2} =(5x+2)^{2}(x^{2} -16)\\441x^{2} =25x^4+20x^3-396x^2-320x-64\\25x^4+20x^3-837x^2-320x-64=0[/tex]
Using a calculator, we have
[tex]x \approx 5.6[/tex] and [tex]x\approx -6.01[/tex].
Therefore, the solution is [tex]x \approx 5.6[/tex].
