Question

in the diagram, tq is 18 units in length. what is the length of rs? 46 units 25 units 18 units 16 units

Explanation:

Step1: Use the property of perpendicular bisector

Since line \(l\) is the perpendicular bisector of \(RQ\), \(SR = SQ\). So, \(2x + 10=9x - 11\).

Step2: Solve the equation for \(x\)

$$
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Step3: Find the length of \(SQ\)

Substitute \(x = 3\) into \(9x-11\), \(9\times3-11=27 - 11=16\).

Step4: Use the Pythagorean theorem in right - triangle \(STQ\)

In right - triangle \(STQ\), \(SQ = 16\), \(TQ=18\). Let \(ST = a\), then \(a=\sqrt{SQ^{2}-TQ^{2}}=\sqrt{16^{2}+18^{2}}=\sqrt{256 + 324}=\sqrt{580}\) (This step is wrong, actually, since \(SR = SQ\) (by the property of perpendicular bisector), we don't need Pythagorean theorem here. The key is that \(SR=SQ\), when \(x = 3\), \(SQ=9x-11=9\times3-11 = 16\) (This is wrong in the previous step explanation, correct as follows)

Since line \(l\) is the perpendicular bisector of \(RQ\), \(SR = SQ\).
Set \(2x + 10=9x - 11\), solve for \(x\):

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Substitute \(x = 3\) into \(9x-11\) (or \(2x + 10\)), \(9\times3-11=16\) (or \(2\times3+10=16\)).

Answer:

16 units