CPhill

First one

ΔABC is similar to ΔLMN by AA congruency

Second one

This is the absolute value funtion shifted 1 unit to the left and 2 units down

The function is    f(x) = l x + 1 l - 2  ⇒  "P"

Lines have infinite length

We should be talking about line segments 

Then the segments have equal length

To see this.......draw a circle of any radius  with center O and any two diameters in the circle

Call the diameters AB   and CD...and these are equal

But  AO = BO = CO = DO  = radii

So....the radii are equal....and since O is the center.......the diameters have the same midpoint

Since  M is a median of AB and  N is the median of AC, their intersection occurs (2/3) of the way from vertex B to N

So... TN     is (1/3) of the distance from B to N =  (1/3)(12) = 4

And  TNC  is a right triangle  with legs NC, TN  and hypotenuse CT

So

CT  =  sqrt ( NC^2 + TN^2 )  =  sqrt ( 7^2 + 4^2) = sqrt ( 65)

Second one

Because VU   splits QS and RS proportionally, then VU is  ll QR is  ll TR

And because VT splits QR and QS proportionally, then VT is  ll RS ll RU

So...TVUR is a parallelogram because opposite sides are parallel

Then   VU = TR  = 6

So....QR  = 2*TR = 2*6  =   12

First one

DE/ DB = AC /AB

DE / DB =  AC / [ 2DB]         multiply through by DB

DE / 1 = AC / 2

DE = AC /2

DE = 12.6 / 2 =   6.3

Inverting the fraction in the denominator and multiplying, we have

(4x^2 + 2x) /  ( x^2 + x - 2)   *    (3x^2 + 10x + 8) / (8x^2 + x)   =

( 4x^2 + 2x) (3x^2 + 10x + 8)

________________________            factor each term

(x^2 + x - 2) (8x^2 + 4x)

(2x) ( 2x + 1) (3x + 4) (x + 2)

_______________________         "cancel" where possible

(x +2) (x - 1) (4x) (2x + 1) 

(3x + 4)

__________   =    the first answer

2(x - 1)

Yeah...I see my error....!!!

You are correct....!!!

The diagonals of a rhombus intersect at right angles

So....the tangent ratio of c1 is

OD / OC   =  sqrt ( 5^2 - 3.9^2) / 3.9   =  sqrt (9.79) / 3.9

d1 can be found thusly

tan d1   = 3.9 / sqrt (9.79)        take the arctan

artan [ 3.9 / sqrt (9.79) ] = d1 ≈ 51.26°

By Euclid.....we have this

Area A * Area C = Area B * Area D

40 * 30  = 25 * Area D

1200 =  25 * Area D     divide both sides by 25

48 = Area D 

Thanks, heureka

Your second way is more elegant.....it translates easily into a circular function...