a) Due to the question asking for a solution using the Triangle Proportionality Theorem, it is assumed that △RQS is similar to the smaller triangle inside it.

Let us define the point at which the line inside △RQS meets line RQ as T, and the point at which that same line meets line RS as V.

Hence, △RTV ~ △RQS.

According to the Triangle Proportionality Theorem:

$\frac{RT}{RQ} = \frac{TV}{QS} = \frac{VR}{SR}$

Hence, we can substitute all values that we know from the diagram:

$\frac{2x - 2}{2x - 2 + 13} = \frac{TV}{17} = \frac{21 - 7}{21}$

After getting this information, all we have to do is simplify and solve the equation:

$\frac{2x - 2}{2x - 2 + 13} = \frac{21 - 7}{21}$

$\frac{2x - 2}{2x + 11} = \frac{14}{21}$

$\frac{2x - 2}{2x + 11} = \frac{2}{3}$

$3(2x - 2) = 2(2x + 11)$

$6x - 6 = 4x + 22$

$6x - 4x = 22 + 6$

$2x = 28$

$x = 14$

To verify this answer, we can plug in the value of x into the equation:

$\frac{2(14) - 2}{2(14) - 2 + 13} = \frac{2}{3}$

$\frac{26}{39} = \frac{2}{3}$

$\frac{2}{3} = \frac{2}{3}$

Hence, after verification, we can get the answer: x = 14.

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b) To find the perimeter of △QRS, all we have to do is to add all the lengths of the sides of △QRS together.

Hence, 2(14) - 2 + 13 + 17 + 21

= 28 - 2 + 13 + 17 + 21

= 77 units

Therefore, the perimeter of △QRS is 77 units.

Hope this helped!

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3 years ago