Respuesta :

calculista
we know that

 in the triangle TQS
applying the Pythagorean theorem
QS²=TS²+TQ²---------> TQ²=QS²-TS²--------> TQ²=18²-9x²-----> equation 1

in the triangle TRS
TS²=TR²+RS²--------------> TR²=TS²-RS²-------> TR²=9x²-144----> equation 2

in the triangle QTR
TQ²=TR²+36-----------> equation 3


I substitute 1 and 2 in 3
18²-9x²=9x²-144+36--------> 18x²-432=0------> x²=24-------> x=√24
x=2√6
TS=3*x------> 3*2√6-----> 6√6
TS=6√6 units

the answer is
TS=6√6 units 
6 is the square root of 6 units

In triangle TQS the length of the TS = [tex]2\sqrt{6}[/tex] units. Therefore the correct option is A).

Given :

  • RS = 12
  • QR = 6
  • TS = 3x
  • Triangle TQS and triangle TRS are right angle triangles.

Solution :

In triangle TQS applying pythagorean theorem.

[tex]\rm QS^2=TS^2+TQ^2[/tex]

[tex]\rm 18^2 = (3x)^2+TQ^2[/tex]

[tex]\rm TQ^2=324-9x^2[/tex]  ---- (1)

In triangle TRS applying pythagorean theorem.

[tex]\rm TR^2+RS^2=TS^2[/tex]

[tex]\rm TR^2+12^2=(3x)^2[/tex]

[tex]\rm TR^2 = -144+9x^2[/tex]  ---- (2)

In triangle TRQ applying pythagorean theorem.

[tex]\rm TQ^2=TR^2+QR^2[/tex]

[tex]\rm TQ^2=TR^2+6^2[/tex] --- (3)

Substitute the value of [tex]\rm TQ^2\;and\;TR^2[/tex] from equation (1) and (2) in equation (3)

[tex]324-9x^2=-144+9x^2+36[/tex]

[tex]432=18x^2[/tex]

[tex]x^2 = 24[/tex]

[tex]x=2\sqrt{6}[/tex] units

In triangle TQS the length of the TS = [tex]2\sqrt{6}[/tex] units. Therefore the correct option is A).

For more information, refer the link given below

https://brainly.com/question/2263981

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