Question

In the diagram the length of segment TR can be represented by 5x-4.

Answer 1

Step $1$

Find the value of x

we know that

TR=RV -------> given problem

in this problem we have

$RV=2x+5\\TR=5x-4$

so

$2x+5=5x-4\\ 5x-2x=5+4\\3x=9\\x=9/3\\x=3\ units$

Step $2$

In the right triangle TRS

Find the length of the side RS

we know that

Applying the Pythagorean Theorem

$TS^{2} =TR^{2}+RS^{2}\\RS^{2}=TS^{2} -TR^{2}$

in this problem we have

$TS=6x-3\\TR=5x-4$

Substitute the value of x

$TS=6*3-3=15\ units\\TR=5*3-4=11\ units$

$RS^{2}=(15)^{2} -(11)^{2}\\RS^{2}=104\ units^2$

Step $3$

In the right triangle RSV

Find the length of the side VS

Applying the Pythagorean Theorem

$VS^{2} =RV^{2}+RS^{2}$

in this problem we have

$RV=2x+5=2*3+5=11\ units\\RS^{2}=104\ units^2$

Substitute in the formula

$VS^{2} =11^{2}+104$

$VS^{2}=225\ units^2$

$VS=15\ units$

therefore

the answer is the option D

the value of the side VS is $15\ units$

Answer 2

Answer: The correct option is D, i.e., 15 units.

Explanation:

It is given that the  length of segment TR can be represented by 5x-4.

From figure it is noticed that the side TR and RV is equal and the length of segment RV is 2x+5. So,

$5x-4=2x+5$

$3x=9$

$x=3$

The value of x is 3, so the length of side RV is,

$2x+5=2(3)+5=11$

In triangle TRS and angle VRS,

TR=VR

$\angle TRS=\angle VRS=90^{\circ}$

RS=RS (common side)

By SAS rule of congruence triangle,

$\triangle TRS\cong\triangle VRS$

Therefore the side TS and VS are congruent sides.

From figure it is noticed that the length of side TS is 6x-3, therefore the length of side VS is also 6x-3.

$VS=6x-3=6(3)-3=15$

Hence, the length of side VS is 15 units and option D is correct.