Question

Please show work i would appreciate it ​

Answer 1

a) x = 14

b) The perimeter of △QRS is 77 units.

Explanation:

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:

`(RT)/(RQ) = (TV)/(QS) = (VR)/(SR)`

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

`(2x - 2)/(2x - 2 + 13) = (TV)/(17) = (21 - 7)/(21)`

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

`(2x - 2)/(2x - 2 + 13) = (21 - 7)/(21)`

`(2x - 2)/(2x + 11) = (14)/(21)`

`(2x - 2)/(2x + 11) = (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:

`(2(14) - 2)/(2(14) - 2 + 13) = (2)/(3)`

`(26)/(39) = (2)/(3)`

`(2)/(3) = (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!