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The translation rule says....

Take the original triangle and move each point  5 units to the left ( ← )   and   4  units down ( ↓ ).

If the perimeter is 26....the length and width sum  is  1/2 of this  =  13

So.....using the Pythagorean Theorem

L    W

12  1   produces a diagonal of √ [12^2 + 1^2]   = √145

11  2      = √[11^2 + 2^2 ]  =  √125

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7   6       ≈ √ [ 7^2 + 6^2 ]  = √85

And this is the shortest diagonal possible for the given conditions

Suppose functions  \(g\)  and  \(f\)  have the properties that  \(g(x)=3f^{-1}(x)\)  and  \(f(x)=\frac{24}{x+3}\) .

For what value of  \(x\)  does  \(g(x)=15\) ?

g(x)  =  3f^-1(x)

                            Since  g(x)  =  15 ,  we can plug in  15  for  g(x) .

15  =  3f^-1(x)

                            Divide both sides of the equation by  3 .

5  =  f^-1(x)

                            Take  f  of both sides...

f(5)  =  x

                            And  f(5)  =  \(\frac{24}{5+3}\)    so....

\(\frac{24}{5+3}\)   =   x

\(\frac{24}{8}\)  =  x

3  =  x

First one

We can use the sine, here

sin (67.4)  = 12 / r        rearrange as

r  =  12  / sin (67.4)  =     13

Second one

cosB  =  16/ 20   =  4/5

sinB  = √ [ 1 - cos^2B]  =√ [ 1 - (4/5)^2]  = √[9/25]  =  3/5

Third one :

tan A  =  a/b

Note that  angle  BAE  =  angle  ECD

And  angle ABE  =  angle  EDC

So....by AA congruency, triangle ABE  is similar to triangle CDE

And CD : AB   =   2: 3

Then the height  of CDE  = CF   is 2/3 height  of  ABE  = FB

So.... CF : FB  =  2 : 3

And angle BFE  = angle BCD

And angle BEF   =  angle BDC

So,again, by AA congruency, triangle BDC  is similar to triangle BEF

But  there are 5 parts  of BC....and  BF  is 3/5 of these

So......BF  is 3/5  of  BC 

So....EF  is 3/5  of DC   =  (3/5) (100)  =    60 (cm)

To avoid getting confused by the apparent right angles, I changed the picture to make BC more slanted looking.

Since alternate interior angles are congruent....

∠CAB  =  ∠ACD    and

∠ABD  =  ∠BDC

So by AA similarity,  △ABE  ~ △CDE

So if  ED = a,  then   EB = 1.5a

Since corresponding angles are congruent....

∠BDC  =  ∠BEF    and

∠BCD  =  ∠BFE

So by AA similarity,  △BCD ~ △BFE

FE / CD  =  BE / BD

FE / 100  =  1.5a / (1.5a + a)

FE   =   100 * 1.5a / (1.5a + a)

FE   =   60    cm

The x coordinate of each original vertex is shifted 3 units to the left   and the y coordinate is shifted 3 units  up

So.... "c"  is the only possibility

thx everyone

Notice that the number of fish  is

[52 + 61 + 87 ]  =  200

Since  800 fish  would be 4 times this....the  number of blue gill would also be 4 times as much  =

4 * 87   =

348

87 / [52+61+87]  x 800 =348 Blue gill fish.

2.   ⇒   RQTS  =  EDCF

So......angle  R  is the image of angle E 

3. ⇒  DE  =  QR

Thank you guys !!1

2)

This parabola opens to the left  because the directrix  lies to the right of the vertex

So  we have the form

-4px  = y^2     ⇒   p   = 4.75

-4(4.75)x = y^2

-19x = y^2       divide both sides by  (-19)

x  =  (-1/19) y^2

1)

y  =  (1/12)x^2   multiply both sides by 12

12y  = x^2

And we have the form

4py  =  x^2

4p  = 12

p  =3

The vertex  is  (0,0) ...the parabola turns upward   

The focus  is  ( 0, 0 + p )   =   (0, 3)

The directrix  is  y  =  -3

Third answer

If I can see this correctly..... believe that "d" is correct

BC  not a tangent because  ABC  =  (180 - 25 - 60 )  =  95° (not equal to 90 )

Thanks so much, CPhill! Yay! 

We have  12 equal sides  in the 4 triangles

Let s  =  the side of each

The total area is     4 (1/2)(s^2)√3/2   =  √3s^2

So....in a larger equilateral triangle.....each  side  will be  12s/3  =  4s   

So....the area of the largwer equilateral triangle  =  

(1/2) (4s)^2 √3/2  =  16s^2√3/4   =  4√3s^2

So....the desired ratio is     √3s^2  / [ 4√3  s^2 ]   =   1 / 4

3.

Multiply both sides by -8

-8x  =  y^2

We have the form

4px  =  y^2     ⇒     -8  = 4p  ⇒     p  =  -2

The vertex of this parabola  is  (0,0)

The focus  is  ( 0 + p , 0 )   =   ( 0 - 2, 0 )  =  (-2, 0)

The directrix  is  x  =  2

Last answer

4.  Vertex  (0, 0 )    Focus  = (2.5, 0)

This parabola opens to the right

The form is

4px  = y^2

p  = 2.5   so

4 (2.5)x  = y^2

10x  =  y^2   divide both sides by 10

x  = (1/10) y^2

That is correct, CPhill! There is no replacement, so that's the answer.

7 ( n + 4)  =  5(n + 8)

7n + 28  = 5n + 40

2n  =  12

n  = 6

We have 

2/11  *  9/10   =       18 / 110   =   9 /55

The Intersecting Chord Theorem  says that the products of the respective chord segments are equal

2 * 15  =  5 * n

30  = 5 * n          divide both sides by 5

6m  =  n

Number of liters consumed  =  3000 / 100 * 6  =  30 * 6  =  180

Total cost  =  Liters onsumed * Price per liter  = 

180 * 1.20   =  $216

Cost per person    =  216 / 4  =  $54

[3,000 / 100] x 6 = 180 - Liters of gas will be used on a 3,000 Km trip.

180 x $1.20 = $216 - total cost of gas over 3,000 Km trip.

$216 / 4 = $54 - share of each one of the 4 people.