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What is your perspective of fast?

Here's an  image :

Bisect angle ACB.....and this will form altitude CE

And AC = BC    And angle ACG  = angle BCG    And  CE is common

So by SAS triangle ACG is congruent to triangle BCG

Then AG = BG  so triangle AGB is isosceles....and because AGB is right.......then angles ABG and BAG  =  45°

And since CEB is right and ABG = 45°....then angle BGE is also = 45°

Then triangle  BGE is also isosceles  with EB = EG

Draw  XY......and by hypotenuse-leg, triangle AYG is congruent to triangle BXG

So......AY  =  BX.....so  XY is parallel to AB

And because BX splits BC equally.......then  CD  =  ED

And triangle CDX  is similar to triangle CEB

And  since CD  is (1/2)  of CE then   DX =  (1/2)EB =  (1/2) EG

And angle XAB = angle YXG  =  45°  and since GDX is right, then angle XGD = 45°

So triangles  GDX and  GEB   are similar

And because triangle GDX is isosceles.....then DX = DG

Thus   DX  = (1/2)EG   and by substitution  DG = (1/2) EG  ⇒ 2DG = EG = EB

So

EB + DG  =  3DG

ED  =  3DG = CD

Add  DG to everything  

ED + DG  = 4DG  = CD + DG

4DG  =  2 EG  =  CG

But  EG =  EB   ....so....

2EB  = CG

But 2EB  = AB

So....  AB =  CG  

1)    Density = Mass / Volume

Mass = Volume x Density

Mass = 8.9  x  6

Mass =53.4 grams 

2)  The experiment must be done at the temperature of 300K.

300K - 273K(Absolute temp) =27 C - The Lab temperature must be very close to this =~300K

This temp. of 27C =[27 x 9/5] + 32 =80.6F. Since the Lab temp. is 55 F, then you must INCREASE the

Lab temp. by: 80.6 - 55 =25.6 =~26 F. 

Because: 26F + 55F =81F = ~27C = 300K.

3)

1 Pound =453.5924  grams

5 x 453.5924 =2.268 x 10^3 grams.

Thank you Alan!

Everything is clear now.

  !

Thanks!

That is a really good way to explain it!! 

Triangles don't really have "lengths" anyway. 

They have bases, heights, sides, etc.

\(\frac{\frac{2}{3}(5*4)}{3+\frac{6}{7}}\)

Is this the correct format?

Let's say x is the mystery number.

So first you have x.

Then you take away 5/7 of x.

\(x-\frac{5}{7}x\)

That equals 30.

\(x-\frac{5}{7}x=30\)

This is the same as 2/7x equals 30.

\(\frac{2}{7}x=30\)

You divide both sides by 2/7.

\(x=\frac{30}{\frac{2}{7}}\)

But when you divide by a fraction, it is the same as multiplying by the reciprocal.

\(x=30*\frac{7}{2}\)

Simplify.

\(x=\frac{30*7}{2}\)

\(x=\frac{210}{2}\)

\(x=105\)

A good way to test this is by plugging it into the original equation.

\(x-\frac{5}{7}x=30\)

Substitute each x for 105.

\(105-\frac{5}{7}(105)=30\)

\(105-\frac{525}{7}=30\)

\(105-75=30\)

\(30=30\)

This is true, so your mystery number is 105.

\(2.[3-2.(2-6)]+1 = ?\)

I do not see how this problem is possible as 2-6=-4 and a decimal cannot be negative.

Questions become dim when they have negative points.

CPhill, you are the best man

The center of the circle is ( -1, 4 )

And the given point is ( 3, 1 )

Finding the slope between these two points, we have 

[  4 - 1 ] / [ -1 - 3]  =  [ 3] / [-4] =  -3/4

And the slope of the tangent ine will be the negative reciprocal of this   = 4/3

So.....the equation of the tangent line is :

y =  (4/3) (x - 3) + 1    simplify

y = (4/3)x - 4 + 1

y = (4/3)x - 3

Here's the graph : https://www.desmos.com/calculator/5u5jc97jd5

All three are real.....!!!!

What is given ???

1.Center on line x=3, tangent y-axis at (0, 5)

The circle  will have its center at  (3, 5)  and since it is tangent to the y axis, the radius will be 3.....so....the equation is

(x - 3)^2  + (y - 5)^2  = 9

Here's the graph : https://www.desmos.com/calculator/sejlwks7kv

2.Radius 4, tangent to x-axis; contain (–5,8)

The center must be (-5, 4).....so the equation is

(x + 5)^2  +  ( y - 4)^2  =  16

Graph here : https://www.desmos.com/calculator/4r1rrwt5pq

3. Radius 5, tangent to y-axis; contain (10,7)

The center is (5,7)   and the radius is 5......the equation  is

(x - 5)^2  + ( y - 7)^2  = 25

https://www.desmos.com/calculator/70z6uvpvh7

4.Center on line y–4=0, tangent x-axis at (–2, 0)

Note......adddig 4 to both sides of the equation produces  y  = 4

So....since it is tangent to the x axis...the radius is 4

And the center is (-2, 4)

So the equation is

(x + 2)^2  +  ( y - 4)^2  =  16

https://www.desmos.com/calculator/edaxudoeg7

You know, blarney master, you really should cut it off, short --just use initials for your name. BM, seems more apropos, anyway. 

What are you trying to do? What do the directions say?

If they are only giving you a portion of the crcle.......you can use (portion) * pi ^ radius^2

For.....example.......find the area of (1/6) of a circle with a radius of 5 ......so we have...

Area  =     (1/6) * pi * (5)^2  ≈   13.1  square units

I don't know if this is what you needed, or not......

When the number is a 1 digit number (1 digit number in base 7)- the numbers 0, 1, 2, 3, 4, 5, and 6

When the number is a 2 digit number=a*7+b=b*16+a / subtract (a+b)

a*6=15*b / divide by 3

2*a=5*b meaning a is divisible by 5 meaning a=5 (because if a=0 then b=0 and then the expression ab is not defined because the digit that represents the largest number in a certain number cannot be 0) meaning b=2. So there is only 1 2 digit number=52 (in base 7)=37 (in base 10) 

When the number is a 3 digit number=49*a+7*b+c=256*c+16*b+a /subtract (a+c+7*b)

48*a=255*c+9*b / divide by 3

16*a=85*c+3*b. c cannot be 0 because it means the expression cba is not defined. if c=1-

16*a=85+3*b. there are no solutions to tht equation. if c>1- this means 16*a=85*c+3*b>=170 meaning a>6. But this is not possible! therefore there is no solution.

So the only numbers are 0, 1, 2, 3, 4, 5, 6 and 37. But Wait! didn't i miss something? what about 4 digit numbers? 5 digit numbers? 12 digit numbers? It is not possible for a number with more than 3 digits to maintain the attribute. How do i know? Well, i have discovered a truly remarkable proof of this theorem which this post is too small to contain. Can you prove it ;)?

(the sum of the numbers is 58)

~blarneymaster~

You can't solve three variables one equation. It usually has infinite solutions. 

Step 1: Write down the decimal divided by 1, like this: decimal 1. Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.) Step 3: Simplify (or reduce) the fraction.

*when the text is dim, not semi-transparent

I took the liberty to add the "=" sign !!!

Solve for x:
x^2 - 18 x - 77 = 0

Add 77 to both sides:
x^2 - 18 x = 77

Add 81 to both sides:
x^2 - 18 x + 81 = 158

Write the left hand side as a square:
(x - 9)^2 = 158

Take the square root of both sides:
x - 9 = sqrt(158) or x - 9 = -sqrt(158)

Add 9 to both sides:
x = 9 + sqrt(158)        or          x = 9 - sqrt(158)

There is no equation here because of the lack of an equal sign. Thus, there cannot be any "solving."