Aanswer (a) and (b) a) a line pq is parallel to the line MN

b) The line MN is four times the length of the line pq​

Step-by-step explanation:

a). From the given picture,

    It's given:  PM = 3(LP) and QN = 3(QN)

    In ΔLPQ and ΔLMN,

    LM = LP + PM

         = LP + 3LP

         = 4(LP)

    Similarly, LN = 4(LQ)

    By the converse property of similar triangles,

    Corresponding sides of the triangles are proportional.  

    Therefore, their angles will be equal in measure.

    ∠LPQ ≅ ∠LMN and ∠LQP ≅ ∠LNM

    Line PQ will be parallel to the line MN (By Converse of corresponding angle theorem).

b). Since, ΔLPQ ~ ΔLMN their corresponding sides will be in the same ratio.

    MN = 4PQ  

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240