Answer:
XT=6 units
Step-by-step explanation:
The picture of the question is the attached figure
step 1
In the right triangle RST
Applying the Pythagorean theorem
[tex]RS^2=RT^2+TS^2[/tex]
we have
[tex]RS=RX+XS=4+9=13\ units[/tex] ---> by segment addition postulate
substitute
[tex]RT^2+TS^2=169[/tex] ----> equation A
step 2
In the right triangle RTX
Applying the Pythagorean theorem
[tex]RT^2=RX^2+XT^2[/tex]
we have
[tex]RX=4\ units[/tex]
substitute
[tex]RT^2=4^2+XT^2[/tex]
[tex]RT^2=16+XT^2[/tex]
[tex]XT^2=RT^2-16[/tex] ----> equation B
step 3
In the right triangle XTS
Applying the Pythagorean theorem
[tex]TS^2=XS^2+XT^2[/tex]
we have
[tex]XS=9\ units[/tex]
substitute
[tex]TS^2=9^2+XT^2[/tex]
[tex]TS^2=81+XT^2[/tex]
[tex]XT^2=TS^2-81[/tex] ----> equation C
step 4
equate equation B and equation C
[tex]TS^2-81=RT^2-16[/tex]
[tex]TS^2-RT^2=81-16[/tex]
[tex]TS^2-RT^2=65[/tex] ----> equation D
step 5
Solve the system
[tex]RT^2+TS^2=169[/tex] ----> equation A
[tex]TS^2-RT^2=65[/tex] ----> equation D
Solve by elimination
Adds equation A and equation D
[tex]RT^2+TS^2=169\\TS^2-RT^2=65\\---------\\TS^2+TS^2=169+65\\2TS^2=234\\TS^2=117[/tex]
Find the value of RT^2
[tex]RT^2+117=169\\RT^2=52[/tex]
step 6
Find the value of XT
equation C
[tex]XT^2=117-81\\XT^2=36\\XT=6\ units[/tex]
