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I forgot to divide by 3, so the distance is 3*sqrt(3).

Sorry, that isn't correct. In case you missed it, the lengths 10 and 24 are the lengths of the diagonals.

The distance from P to line AC is sqrt(18^2 - 9^2) = 9*sqrt(3).

Actually it was 32.

20234-19876  ?   Assuming no leading 0 allowed

It's 20 trust me

thank you!

2. In triangle ABC, the acute angles are angle B=60 and angle C=30. The centroid of triangle ABC is G If AB=1, what is AG? (no picture)

Notice that

\(\displaystyle (x+\sqrt{xy}+y)(x-\sqrt{xy}+y)=x^{2}+xy+y^{2},\\ \text{ so }\\9(x-\sqrt{xy}+y)=27,\\ \text{and therefore}\\ x-\sqrt{xy}+y = 3.\)

Here is an approximate diagram.

You can get exact figures using trigonometry.

1. DEF is the medial triangle of ABC and XYZ s the medial triangle of DEF. If the perimeter of XYZ is 5 what is the perimeter of ABC? (no picture)

1. DEF ist das mediale Dreieck von ABC und XYZ ist das mediale Dreieck von DEF. Wenn der Umfang von XYZ 5 ist, was ist der Umfang von ABC? (kein Bild)

Hello Guest!

\(P_{ABC}=2\times P_{DEF}\\ P_{DEF}=2\times P_{XYZ}\\ \color{BrickRed}P_{XYZ}=5 \)

\(P_{ABC}=5\times 2\times 2\)

\(P_{ABC}=20\)

  !

The positive integers from 1 to 150 inclusive are placed in a 10 by 15 grid so that each cell contains exactly one integer. Then the multiples of 3 are given a red mark, the multiples of 5 are given a blue mark, and the multiples of 7 are given a green mark. How many cells have more than 1 mark?

This diagram is just to get you started.  Work out the very middle overlap section first. Then work out from there.

HINT: The numbers that have all 3 colours must have 3*5*7 =105 as a factor.  

Please no one give a full answer.

I do not understand the question

There is a 6-digit sequence of 8 positive consecutive integers that divides 2, 3, 5, 7, 9, 11, 13 and 17 EVENLY. What is the sequence in question? Thank you for help.

What does this mean....There is a 6-digit sequence of 8 positive consecutive integers     ??

2x  -  y   =    a   ⇒   6x - 3y  =  3a   (1)

3y - 6x    =  b        (2)

Adding (1)  and (2)   we get that

0   =  3a +  b

-3a  =  b

So

a / b   =

a  /  (-3a  )   =

- 1/ 3

If x + sqrt(xy) + y = 9 and x^2 + xy + y^2 = 27, find x - sqrt(xy) + y.

x + sqrt (xy) + y  =  9

x + y  =  9 - sqrt (xy)           square both sides

x^2 + 2xy + y^2  =  81 - 18sqrt (xy) + xy     simplify

x^2 +xy + y^2  =  81 -  18 sqrt (xy)

27  =  81 - 81sqrt (xy)

-54  = -18 sqrt (xy)

3  =  sqrt (xy)

And

x + sqrt (xy) + y  =  9

x + (3 + y)  = 9

x + y  = 6

So

x - sqrt (xy) + y  =

x + y - sqrt (xy)  =

6   - 3   =   

3

OK, GM....sorry for some of the silly errors   !!!

This is a job for a computer code to find! How else can you test 10,000,000 numbers for a sequence of 8 consecutive integers? So, here is your computer code to do just that.

a=1; b=10000000;c=if(a%2==0 and (a+1)%3==0 and (a+2)%5==0 and (a+3) %7==0 and (a+4)%9 and (a+5)%11==0 and (a+6)%13==0 and (a+7)%17==0, goto loop, goto next); loop:printa," ",;next: a++;if(a

OUTPUT- There is no 6-digit sequence that the computer could find !!  However, it found a 5-digit sequence of 8 integers as follows:(67568, 67569, 67570, 67571, 67572, 67573, 67574, 67575)=67568 mod 2=0, 67569 mod 3=0, 67570 mod 5=0, 67571 mod 7 =0, 67572 mod 9 =0, 67573 mod 11 =0, 67574 mod 13 =0, 67575 mod 17 =0.

P. S. It also found 6 different sequences each made up of 7-digit integers as follows:
(1599098,  3130628,  4662158,  6193688,  7725218,  9256748) - Each of these sequences begins with the first term and each subsequent term is incremented by 1: Example: 1599098 , 1599099, 1599100, 1599101, 1599102, 1599103, 1599104, 1599105.

6 (a)

Here's the graph https://www.desmos.com/calculator/ej36jt2v3h

(b)  We actually have a trapezoid  here

The height  = 4

One base  =  20

The other  = 180

So...the area (above the x axis )  is

(1/2) (height) (sum of bases)  = 

(1/2) (4) ( 20 + 180)  =

(1/2) (4) ( 200)  =

2 (200)  =

400 units^2

(c)  The function is a velocity function.....the area under the  curve represents the total distance traveled between hour 0  and hour 4

When an orchestra plays a national anthem, its musicians are ordered in a square. When the orchestra plays any other song, the musicians are ordered in a rectangle such that the number of rows increases by five. What is the number m of musicians in the orchestra?  

There are 400 musicians in the orchestra. 

For the National Anthem, they stand in a 20 x 20 square. 

For the other songs, they rearrange to a 25 x 16 rectangle. 

_.

Fixed....it doesn't affect anything

If   x^2   existed  at x  = 1    then  y  = 1

At x = 1  then the yvalue for 2x - 5  =  -3

So...as the graph shows  here : https://www.desmos.com/calculator/vydmkdw06y, it is not continuous...

You are correct, GM....just a simple mistake...let me fix that....!!!

Since AB  = AD, let  angles ADB  and ABD  =  x

And  let  angle DAB   =  angle CAB  =  z

Let angle DBC   = y

And let angle  ABC  =  x+ y

Andl let angle  ACB  =  x + y  - 32

So...we have the following equations :

(x + y ) + [ (x + y)  - 32]  + z   = 180   ⇒    2x + 2y  + z  = 212  ⇒   (2x + z)  + 2y  =  212     (1)

2x + z  = 180     (2)

Sub (2)  into (1)   and we have that

180 + 2y = 212

2y = 32

y  = 16°   = angle  DBC