Question

What are expressions for MN and LN? Hint Construct the altitude from M to LN.

MN = 1

(Type an exact answer, using radicals as needed.)

Answer 1

The question is missing the figure. So, it is in the atachment.

Answer: MN = x
`√(2)` LN =
`(x)/(2).(√(2) + √(6) )`

Step-by-step explanation: The first figure in the attachment is the figure of the question. The second figure is a way to respond this question by tracing the altitude from M to LN as suggested. When an altitude is drawn, it forms a 90° angle with the base, as shown in the drawing. To determine the other angle, you have to remember that all internal angles of a triangle sums up to 180°.

For the triangle on the left of the altitude:

45+90+angle=180

angle = 45

For the triangle on the right:

30+90+angle=180

angle = 60

With the angles, use the Law of Sines, which is relates sides and angles, as follows:

`(a)/(sinA) = (b)/(sinB) = (c)/(sinC)`

For MN:

`(x)/(sin(30)) = (MN)/(sin(45))`

MN =
`(x.sen(45))/(sen(30))`

MN = x
`√(2)`

For LN:

`(LN)/(sen(105)) =(x)/(sin(30))`

LN =
`(x.sin(105))/(sin(30))`

We can determine sin (105) as:

sin(105) = sin(45+60)

sin(105) = sin(45)cos(60) + cos(45)sin(60)

sin(105) =
`(√(2) )/(2).(1)/(2) + (√(2) )/(2).(√(3) )/(2)`

sin(105) =
`(√(2) )/(4) + (√(6) )/(4)`

LN =
`(x.sin(105))/(sin(30))`

LN =
`x.((√(2) )/(4) + (√(6) )/(4) ) .2`

LN =
`(x)/(2).(√(2) + √(6) )`

The expressions for:

MN = x
`√(2)`

LN =
`(x)/(2).(√(2) + √(6) )`