Answer: The value of k is 2.
Explanation:
From the figure it is noticed that the triangle LMN, LON and MNO are right angle triangle.
According to the pythagoras theorem,
[tex](hypotenuse)^2=(base)^2+(perpendicular)^2[/tex]
Use pythagoras theorem in triangle LON.
[tex](LN)^2=(ON)^2+(LO)^2[/tex]
[tex]m^2=(4)^2+(8)^2[/tex]
[tex]m^2=80[/tex]
Use pythagoras theorem in triangle MON.
[tex](MN)^2=(MO)^2+(NO)^2[/tex]
[tex]l^2=k^2+(4)^2[/tex]
[tex]l^2=k^2+16[/tex] ...... (1)
Use pythagoras theorem in triangle LMN.
[tex](LM)^2=(MN)^2+(LN)^2[/tex]
[tex](8+k)^2=l^2+m^2[/tex]
[tex](8+k)^2=l^2+80[/tex]
[tex](8+k)^2-80=l^2[/tex] .... (2)
From equation (1) and (2), we get
[tex]k^2+16=(8+k)^2-80[/tex]
[tex]k^2+16=k^2+16k+64-80[/tex]
[tex]16=16k-16[/tex]
[tex]32=16k[/tex]
[tex]k=2[/tex]
Therefore the value of k is 2.
