Bosco

Let X be a point inside rectangle ABCD.  If AX = 3, BX = 3, and CX = 3, then find DX.   

                                        AX² + CX²  =  BX² + DX²         

                                         3²  +  3²  =  3²  +  DX²     

                                         DX²  =  3²  

                                          DX  =  3    

i copied the following from the internet.  

In Euclidean geometry, the British Flag Theorem says that if a point P is chosen inside a rectangle ABCD, then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners. 
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Let "r" = the amount added to each successive term.  

a₁ + r = a₂   and   a₂ + r = a3   therefore a₃ = a₁ + r + r    

given that a₁ + a₃ = 5   therefore (a₁) + (a₁ + 2r) = 5                 (eq 1)  

a₁ + r = a₂   and   a₂ + r = a₃   and   a₃ + r = a₄   therefore a₄ = a₁ + r + r + r   

given that a₂ + a₄ = 6   therefore (a₁ + r) + (a₁ + 3r) = 6            (eq 2)   

subtract (eq 2) minus (eq 1)   

(eq 2) ...............................................  2a₁ + 4r = 6    

(eq 1) ...............................................  2a₁ + 2r = 5    

                                                                     2r = 1         

                                                                       r = 0.5      

substitute r back into (eq 2)               2a₁ + (4)(0.5) = 6   

                                                                       2a₁ = 4  

                                                                         a₁ = 2     

check answer  

                          a₁ = 2.0 , a₂ = 2.5 , a₃ = 3.0 , a₄ = 3.5    

                          a₁ +  a₃ = 5                a₂ +  a₄  = 6   

                         2.0 + 3.0 = 5               2.5 + 3.5 = 6    

Let  
f(x) = \sqrt{x}  
Find the largest three-digit value of x such that f(x) is an integer.  

31²  is    961   

32²  is  1024   

Therefore, x = 961   

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In any polygon, the sum of the interior angle and the exterior angle at a vertex is 180^o.  

In this polygon, the value of the interior angle is twice that of the exterior angle,    

therefore, the interior angle is 120^o and the exterior angle is 60^o.   .  

In any regular polygon, the sum of all the exterior angles is 360^o     

irrespective of the number of vertexes.  

Since one exterior angle of this polygon is 60^o     

there must be six of them to total of 360^o. 

We shall presume that there are the same number of sides that there are vertexes.   

Therefore, this polygon has six sides.    

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  2x^2 + 3x + 8x - x^2 + 4x + k has a double root    

  combine like terms                          (2x² – x²) + (3x +8x + 4x) + k  

                                                           x² + 11x + k    

  to have a double root,    

  i.e., both roots are the same,   

  the discriminate must equal zero        b² – 4ac = 0    

                                                            (11)² – (4)(1)(k)  =  0    

                                                             121 – 4k  =  0    

                                                                     –4k  =  –121    

                                                                          k  =  121 / 4    

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1.   At the vertex of a polygon, the sum of the exterior angle and the interior angle is 180^o.   

      Since the interior angle is 12 times the exterior angle, the exterior angle has to be 15^o.  

      The sum of all exterior angles of a polygon is 360^o, irrespective of the number of sides.  

       Therefore, the polygon has 360 / 15 = 24 vertices.   

       We shall presuppose that a polygon has the same number of sides as it has vertices,    

       Therefore, this polygon has 24 sides.    

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                                       y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25    

group like terms              y  =  (–2x² – 3x²) + (8x – 14x) + (–15 + 25)    

combine like terms           y  =  –5x² – 6x + 10    

first derivitive                    ^dy/_dx  =  –10x – 6    

set equal to zero                    0  =  –10x – 6    

                                           10x  =  –6   

                                               x  =  – 0.6    this is the x-coordinate of the vertex

sub x into original                   y  =  [ –5 • (–0.6)² ]  –  [6 • (–0.6) ]  +  10    

                                               y  =  (–5 • 0.36)  –  (–3.6)  +  10    

                                               y  =  –1.8 + 3.6 + 10    

                                               y  =  11.8      this is the y-coordinate of the vertex

the vertex is at                        (–0.6 , 11.8)    

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                                                 x^2-mx+24 = 10   

subtract 10 from both sides      x² – mx + 14  =  0   

integers that multiply to 14        (+7)(+2)   

                                                  (–7)(–2)   

                                                  (+1)(+14)   

                                                  (–1)(–14)  

possible values of m                   –9   

                                                   +9   

                                                  –15   

                                                  +15   

total number of

possible values of m                      4    

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                                        5x^2 - 2x + 8 - 2x^2 + 6x + 2 - 2x^2 + 4x - 1   

can be regarded as        (5x² – 2x² – 2x²) + (–2x + 6x + 4x) + (8 + 2 – 1)   

combine like terms            x² + 8x + 9   

discriminant is                   8² – (4)(1)(9)  =  64 – 36  =  28   

The discriminant is the expression under the radical   

in the quadratic formula.   

When a quadratic is stated in the form ax² + bx + c  

the discriminant is b² – 4ac.  

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