Melody

I have now answered on yesterdays post. 

I have to continue on  a second post because it is not displaying properly.

\( \text{Now consider the points MPN, these points either form a triangle or they are collinear}\\ \text{If they form a line then MP+PN=MN}\\ \text{If they from a triangle then use this fact:}\\ \text{In any triangle the sum of any 2 side lengths must be longer than the third side.}\\ So\;\; \\ {MP}+{PN} \ge{MN}\\ 2 {MP}+2{PN}\ge2{MN}\\ AB+CD\ge2MN\\ 2MN\le AB+CD\\ MN \le \frac{AB+CD}{2} \)

There can only be an equality if  P lies on the line MN in which case there is no triagle MPN and the the quadrilateral ABCD must be a trapezium (Australian definition, I think Americans call it a trapezoid) where  \(AB \| DC\)

\(Consider\;\; \triangle BDC\;\;and\;\;\triangle BPN\\ \angle PBN=\angle DBC \;\qquad common\;\;angle\\ \frac{\overline{ BD}}{\overline{BP}}=\frac{\overline{ BC}}{\overline{BN}}=2\\ \therefore \triangle BPN \sim \triangle BDC\\ \qquad      \text{One equal angle and leg lengths from it in the same ratio.}\\ So \;\;\overline{DC}=2 \cdot \overline{PN}      \qquad \text{ corresponding sides in similar triangles with a dilation factor of 2}\)

Using the same logic with \(\triangle APD\;\; and\;\; \triangle MPD\) it can be shown that  

\(\overline{AB}=2\cdot \overline{MP} \)

I assume you mean

if its spring now what was it 3 months ago

In Australia it is clearly defined.

Winter comes before spring.

So the answer is Winter.

Our seasons are defined by months, I think in the US it works differently.

Hint:

What multiplies to give  -8

It is negative so one is neg and one is positive.

1, -8

-8,1

-4,2

-2,4

They are the only ones.  which 2 add to give 7?

Ok thanks. 

Do you know why ?

If so would you care to share your knowledge?

Thanks Ginger  

Perhaps you should tell the other guest how you got your answer.

Maybe the person does not know what perimeter or area even mean. 

Here is the formula for the surface are of a cone.

Just plug the values in that you know and solve for h

then us pythagoras and solve for l  (slant height)

Hi Ginger,

5 of the 9 letters of the word MINCEMEAT are selected.

Find the number of possible selections which contain exactly 1 M and 1 E.

I just mean that if 5 letters are selected at random then the prob of drawing MECAT is higher then the prob of drawing INCAT    (becasue there are 2Ms and 2Es to chose from)

So saying there are 10 distinct outcomes is correct but each one  does not have the same probability of occuring as other possible outcomes.

eg

What is the probability of chosing a selection of 5 letters that contain exactly one M and one E

We have determined that there are 10 such unique combinations

How many combinations (not unique) are there altogether.  (think in terms of  M1, M2, E1, E2)

9C5 = 126

Now how many of these have exactly one M and one E

M1 OR M2,  E1 or E2, plus 3 more.

2*2*5C3 = 40

So I think that if 5 letters are selected at random then the prob of having exactly one M and one E is     40/126 = 20/63 

I've played with this a little and Rom's answer is most likely correct but I do not see why.

If I spent hours on proofs I am sure I could show it but I don't see it intuitively at all.

I was surprised when that angle came up as 77 degrees.

Thanks Guest, now the first one makes total sense.

There are 3 to chose from out of 5 then add in the M and the E.

So, yea that is the correct number of possible arrangements, I'm in total agreement.

For the second question it is saying you can have

ME then chose 3 out of the 5       5C3       or

MEE then chose 2 out of the 5      5C2     or

MME then chose 2 out of the 5      5C2     or

MMEE then chose 1 out of the 5     5C1  

Add those all up and you have the number of ways.

Yes that makes total sense too. I just realized that this second answer is exactly what I did in the first place LOL.

I cannot see any scenario where 10*5C3 would be the answer to the first one.

I do not understand what you have written anyway. There is only 2 questions. What are all those combinations in the bracket for?