Answer:
x = 15
Step-by-step explanation:
ΔPQR and ΔSQT are similar (shown by the ~ symbol), so all pairs of corresponding sides have the same scale factor. That means this:
ST and PR are corresponding. QT and QR are also corresponding. Given that information, you can be sure that:
[tex]\frac{PR}{ST}=\frac{QR}{QT}[/tex]
Using that information, you can solve this problem. In the given triangles, there are 2 pairs of corresponding sides that have measurements. ST and PR are both measured, and QS and QP are measured. In this case, the length of QP is actually the sum of SQ and SP, so [tex]x+13[/tex].
[tex]\frac{PR}{ST}=\frac{QP}{QS}\\\\\frac{21}{x-9}=\frac{x+13}{8}\\\\21\times8=(x-9)(x+13)\\168=x^2+13x-9x-117\\168=x^2+4x-117\\x^2+4x-285=0[/tex]
At this point, you need to factor to find x. With an equation in the form of:
[tex]ax^2+bx+c[/tex]
you need to find 2 numbers that multiply to make ac and add to make b. Here, we need 2 numbers that multiply to -285 and add to make 4. I won't write it all out here, but it is very helpful to factor -285 and check each pair to see which adds to 4.
In this case, those 2 numbers are -15 and 19. Using those 2 numbers, you can write the factored form.
[tex]x^2+4x-285=0\implies(x-15)(x+19)=0[/tex]
Finally, you can solve for x. What 2 values of x make this equation true? Looking in the first pair of parenthesis, [tex]x=15[/tex] would evaluate to zero, and that would make the equation true. In the second pair, [tex]x=-19[/tex] would also make the equation true.
[tex]x=15,-19[/tex]
Because we can't have a negative side length, the answer is just x = 15.
