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All the sides of a triangle are integers.  If the perimeter of the triangle is $3,$ then how many different possible triangles are there?  (Assume that the triangle is non-degenerate.  Two triangles are considered the same if they are congruent.)    

Only one possible triangle with perimeter three ... its integer sides are 1 - 1 - 1.     

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There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.   

Little Village ----- 4 ----- Your Town ----- 1 ----- Capital City ----- 2 ----- My Town   

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This just means that the discriminant is = 0    ( double root) 

x^2 + (k-9) x +16  = 0 

   b^2 - 4ac = 0 

   (k-9)^2 - 4(1)(16) 

    k^2 - 18k + 81 - 64 = 0 

     k^2 - 18k + 17 = 0                     Use quadratic formula to find k = 1 or  17

Another way: 

     if     (1/2(k-9)) ^2 = 16   this will be    ( via 'completing the square' for 'x' ) 

               1/2k - 4.5  = +- 4

                  1/2 k =   .5  or 8.5 

                     k =  1  or 17 

The volume of the cone is 200*pi.

The volume of the sphere is 500*pi.

The surface area of the sphere is 200*pi.

The volume of the pyramid is 30*sqrt(3).

The volume of the rectangular prism is 1000.

X^2 + Y^2 = 1      IS THE UNIT CIRCLE   with center at the origin ...this has a RADIUS of  1

   the line y = x passes through the origin 

      distance between A and B  will be   :   2  unit

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Let's denote the area of the shaded region as S.

Observation: Notice that the shaded region consists of 4 smaller quadrilaterals: APWV, PQRS, RTUC, and UVWD. Each of these smaller quadrilaterals is similar to the original quadrilateral ABCD.

Ratio of Areas: Since the sides of the smaller quadrilaterals are one-third the length of the corresponding sides of ABCD, their areas will be one-ninth the area of ABCD.

Calculating the Area of the Shaded Region: Therefore, the area of each smaller quadrilateral is 180/9=20.

So, the total area of the shaded region, S, is:

S=4∗20=80

Therefore, the area of the shaded region is 80.

Find all values of a that satisfy the equation

\(\dfrac{a}{3} + 1 = \dfrac{a + 3}{a} - \dfrac{a^2 + 2}{a}\\ \dfrac{a^2}{3}+a=a+3-a^2-2\\ \dfrac{4}{3}a^2=1\\ a^2=\dfrac{3}{4}\)

\(a=\dfrac{\pm \sqrt{3}}{2}\\ \color{blue}a\in \{-\dfrac{\sqrt{3}}{2},\dfrac{\sqrt{3}}{2}\}\)

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\(\overline{PM}=\color{blue}\dfrac{\overline{PQ}+\overline{PR}}{2}\)

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https://web2.0calc.com/questions/triangles_143

There is a group of five children, where two of the children are twins. In how many ways can I distribute $3$ identical pieces of candy to the children, if the twins must get an equal amount of candy?     

First, give the twins each one piece of candy.       

That leaves three kids and one piece of candy.     

There are 3 ways you could give that last piece.    

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