hectictar

By definition of tangent,

tan θ  =  sin θ / cos θ

                                         Plug in  1/5  for  cos θ , and  2√6  for  tan θ .

2√6  =  sin θ / (1/5)

                                         Multiply both sides of the equation by  1/5 .

(1/5)(2√6)  =  sin θ

sin θ  =  2√6 / 5               

5x - 17  =  -x + 7

Add  x  to both sides of the equation.

6x - 17  =  7

Add  17  to both sides of the equation.

6x  =  24

Divide both sides by  6 .

x  =  4

Check:  does   5(4) - 17  =  -4 + 7   ?

                                   3   =   3          Yep!

Let  x  be the side length of the triangle. Here's a quick drawing:

From the Pythagorean theorem...

(x/2)² + (√6)²  =  x²

x²/4 + 6  =  x²                Multiply through by  4 .

x² + 24  =  4x²               Subtract  x²  from both sides.

24  =  3x²                       Divide both sides by  3  then take the positive square root of both sides.

x  =  √8

And...

area of triangle, in square units  =  (1/2)(x)(√6)   =   (1/2)(√8)(√6)   =   (1/2)(4√3)   =   2√3

Let's call the length of each leg of the right triangle  " t " .

And let's call the side length of the square  " s " .

The triangle is an isosceles right triangle, so

t² + t²  =  s²

2t²  =  s²

And...

area of triangle  =  (1/2)(t * t)   =  t²/2

area of pentagon  =  area of square + area of triangle

area of pentagon  =          s²             +         t²/2                Substitute  2t²  in for  s² .

area of pentagon  =          2t²            +         t²/2

So...

\(\frac{\text{area of triangle}}{\text{area of pentagon}}=\frac{\frac{t^2}{2}}{2t^2+\frac{t^2}{2}}=\frac{\frac12}{2+\frac12}=\frac12\cdot\frac25=\frac15=\frac{20}{100}\)

The area of the triangle is  20%  of the area of the pentagon.  

     \(\frac{1}{1+\frac{1}{\sqrt2}}\)

                     Rewrite the  1  in the denominator as  \(\frac{\sqrt2}{\sqrt2}\) .

=  \(\frac{1}{\frac{\sqrt2}{\sqrt2}+\frac{1}{\sqrt2}}\)

                     Add the fractions in the denominator together.

=  \(\frac{1}{\frac{\sqrt2+1}{\sqrt2}}\)

                                That is the same as...

=  \(1\div\frac{\sqrt2+1}{\sqrt2}\)

                                Invert the second fraction and multiply.

=  \(1\,\cdot\,\frac{\sqrt2}{\sqrt2+1}\)

=  \(\frac{\sqrt2}{\sqrt2+1}\)

                                Multiply the numerator and denominator by  \(\sqrt2-1\) .

=  \(\frac{\sqrt2}{\sqrt2+1}\,\cdot\,\frac{\sqrt2-1}{\sqrt2-1}\)

=  \(\frac{2-\sqrt2}{2-1}\)

=  \(\frac{2-\sqrt2}{1}\)

=  \(2-\sqrt2\)

Assuming that the equation is this...

(x + 6) / (x - 3)  =  2x / (x - 3)

Any  x  value that causes a zero in a denominator is excluded from the domain.

So, we can't have

x - 3  =  0

x  =  3         This is the only value that is restricted.

The two fractions already have a common denominator of  x - 3  ,  so the LCD is just  x - 3 .

(x + 6) / (x - 3)  =  2x / (x - 3)               Multiply both sides by  x - 3 .

x + 6  =  2x        Subtract  2x  from both sides, then multiply both sides by  -1 .

x - 6  =  0           I think this is the standard form.

To solve for  x , add  6  to both sides.

x  =  6                And  6  is not a restricted value.  x = 6  is the only solution.

I don't know what your  " a "  and  " b "  stand for, but it looks like you want the equation of a circle with its center at  (1, 4)  that passes through  (1, 13) .

the radius of the circle  =  the distance between the center and a point on the circle

the radius of the circle  =  the distance between  (1, 4)  and  (1, 13)

Notice how the (1, 4)  and  (1, 13)  both have the same  x  coordinate.

So the distance between the two points is just  13 - 4 .

the radius of the circle  =  13 - 4   =   9

So we know that  r = 9  ,  h = 1 ,  and  k = 4 .

Plugging these values directly into the standard form of a circle.....

(x - h)² + (y - k)²  =  r²

(x - 1)² + (y - 4)²  =  9²

The pie store is having a 20% off sale on all of its pies.

If the pie you want regularly costs $18, how much would you save with the discount?

You'd save  20%  of  $18 .

20%  of  $18   =   0.2 * $18   =   $3.60

11 2/3   =   35/3   =   280/24

7 7/8   =   63/8   =   189/24

The GCF of  280  and  189  is  7 , so....

We can divide the side that is  280/24 m long into  40  equal parts, each  7/24 m long.

We can divide the side that is  189/24 m long into  27  equal parts, each  7/24 m  long.

So the maximum size of a square tile that will fit the room exactly is  7/24 m  by  7/24 m  .