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3.  Call the center of the circle, O....Let M be the mid-point of chord AB            

AO =  r

PAO  is a right triangle  with PAO  =  90°

PO  = sqrt [ AO^2  + PA^2]  = sqrt [ 2^2 + 4^2 ] = sqrt [ 20] =  2sqrt 5

Triangles

AOP  and  MOA  are similar  right triangles

OA / AP  = OM / MA

2/4  = OM / MA

1/2 = OM/ MA OM = 2MA

AO^2   =  OM^2  + MA^2

2^2  = (2MA)^2  + MA^2

4 = 5 MA^2

MA  =  2/sqrt 5

2MA =  AB  =  4 / sqrt 5  =   (4/5)sqrt 5

My square patio is tiled with square tiles, all the same size.  All the tiles are gray, except the tiles along the two diagonals, which are all yellow.  (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.)  If there are $20$ yellow tiles, how many gray tiles are there?  

I can make it work for 21 yellow tiles, but not 20.  

   Y   G   G   G   G   G   G   G   G   G   Y  

   G   Y   G   G   G   G   G   G   G   Y   G 

   G   G   Y   G   G   G   G   G   Y   G   G 

   G   G   G   Y   G   G   G   Y   G   G   G 

   G   G   G   G   Y   G   Y   G   G   G   G 

   G   G   G   G   G   Y   G   G   G   G   G 

   G   G   G   G   Y   G   Y   G   G   G   G 

   G   G   G   Y   G   G   G   Y   G   G   G 

   G   G   Y   G   G   G   G   G   Y   G   G 

   G   Y   G   G   G   G   G   G   G   Y   G 

   Y   G   G   G   G   G   G   G   G   G   Y  

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\( \log_{8a} b \)

=       log b / log (8a)  =   log b  /  [ log 8  + log a ]

2. 

minor arc TA = 2 *angle QTA =   2(47) =  94° 

minor arc UA =  2* angle  RUA  =   2 (65) = 130°      

arc TAU  =  94 + 130 =   224°

minor arc TU  = 360 - 224  = 136°

angle QXR  =  (1/2)  [ arc TAU  - arc TU ]  = (1/2) [ 224 - 136]  = 88° 

I agree - 2^2 is 4, by the way

\(\log_2(\log_3x)=\log_3(\log_2x) \rightarrow \text{ apply change of base to } \log_3 \text{ on both sides and change to } \log_2 \text{.}\)

\(\log_2(\frac{\log_2x}{\log_23})=\frac{\log_2(\log_2x)}{\log_23}\)

\(\log_2(\log_2x) - \log_2({\log_23})=\frac{\log_2(\log_2x)}{\log_23}\)

\(\log_2(\log_2x) \rightarrow z \; \log_23 \rightarrow y\)

\(z-\log_2y=\frac{z}{y}\)

\(z=\frac{y\log_2y}{y-1}\)

\(x=2^{2^{\frac{(\log_23)(\log_2(\log_23))}{\log_2{3}-1}}}\)

for \(\frac{a}{|a|}\) we have 1 if a>0, -1, if a <0. This is easy to see because | | either return the positive version of the number or the negative version of the number.

If all three of x, y, z are positive, then xyz>0, then all four terms are one, producing 4.

If 2 of x, y, z are positive, then xyz<0 then there are two ones and two negative ones resulting in 0.

If one of x, y, z is positive, then xyz>0, and there are two ones and two negative ones, resulting in 0.

If x, y, z are all negative, then xyz<0, and we get all four terms as negative one resulting in -4.

So, all possible values are -4, 0, 4.

Let $ABCD$ be a square with side length $1.$ A laser is located at vertex $A,$ which fires a laser beam at point $X$ on side $\overline{BC},$ such that $BX = \frac{1}{2}.$  The beam reflects off the sides of the square, until it ends up at another vertex; at this point, the beam will stop.  Find the length of the total path of the laser beam.  

The beam starts at corner A and bounces off the middle of side BC.   

The next place the reflected beam should strike would be corner D. 

The beam makes a right triangle ABX with AX being the hypotenuse. 

Using Pythagoras Theorem,                                   AX²  =  1² + 0.5²  

                                                                               AX²  =  1.5  

                                                                                 AX  =  1.118   

The reflected beam would be the same length.  

So, totalling AX + XD,

the total length of the beam is                  1.118 + 1.118  =  2.236    

_.

The solid formed by rotating triangle ABC around side BC is a cone. Here's how to find its volume:

Identify the cone's dimensions:

Radius (r): During the rotation, side AC traces out the base of the cone

 Since AC = 5, the base radius (r) of the cone is 5/2 (half the length of AC becomes the radius).

Height (h): As the triangle rotates, side AB acts as the height of the cone. Therefore, the height (h) of the cone is 5 (AB length).

Formula for Cone Volume: The volume of a cone is calculated using the formula: Volume = (1/3) * π * r^2 * h

Substitute Values and Solve: Volume = (1/3) * π * [(5/2)^2] * 5 Volume = (1/3) * π * (25/4) * 5 Volume = (125/12) * π

Answer: The volume of the solid is (125/12)π cubic units.

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

The length of the median is equal to the average of the two bases.  

So, with one base 12, and the other base 16 longer, we have  

                  12 + (12 + 16)        40  

median  =  ––––––––––––  =  ––––  =  20  

                             2                   2  

The short base has to be 12, because 12 can't be 16 greater than another positive number. 

_.

A cylinder has a height of 6 centimeters and a diameter of 20 centimeters. What is its volume? Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.  

The volume of a cylinder is the area of the base times the height.  

Area of base:                 π r²  =  3.14 • 10²  =  314  

Height:                                                                  6  

Volume:                                                           1884 cm³    

_.

A

             D       12

B                 C

Triangle ABC is  a 45-45-90 right triangle

AB = BC  =   12/sqrt 2 =  6sqrt 2

AD = AD

BD = BD

AB = BC

So triangles  ABD  and CBD  are congruent

Then [ ABD ]   =  (1/2) [ABC]   =  (1/2) ( 1/2) ( 6sqrt 2)^2  =  (1/4) (72)   = 18

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.  

          It could work like this.  Let each x represent one mile.    

           x          x          x          x          x          x          x          x         

          MT                  CC       YT                                          LV      

_.

See a similar problem with heureka's answer here

https://web2.0calc.com/questions/geometry_136

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.)  

A triangle has 3 sides.  

All the way around is 3.  

Every side is an integer.  

Each side can equal only 1.  There is one triangle.  

_.

Mytown  2  Capital City  1 Yourtown 4  Little Village

Capital City - Yourtown =  1 mile

6 and 3 + 2sqrt(2)

x^2  - 6x + 13  =  0

uv = 13

(uv)^2  = u^2v^2  = 169

u + v =  6    square both sides

u^2 + 2uv + v^2   = 36

u^2 + 2(13) + v^2  = 36

u^2 + v^2  =  10

u^3  + v^3 = ( u + v) ( u^2 + v^2  - uv) =   (6) ( 10 - 13) =  -18

u / v^2  + v / u^2  = 

[ u^3 + v^3 ] / [ u^2v^2 ]  =

-18  / 169

2x  -4y + 3z  = 10

4x - 9y - 14z = 7

kx + 5y + z  =  3

The system will have no solution  when  the matrix discriminant  =   0

2    -4     3      2      -4

4   -9     -14    4      -9

k    5      1      k        5

[ 2*-9 * 1   + -4*-14 * k + 3*4*5 ] - [ k* -9 * 3 + 5 *-14*2 + 1*4 *-4 ]   =   0

[-18 + 56k + 60 ] - [ -27k - 140 - 16 ]  =  0

83k + 198  = 0

k = -198 / 83

3x^2 - 16x + 17  =  0

Product of the roots =   17/3  = ab

(ab)^2 = a^2*b^2  = 289 /  9

2ab =  34/3

Sum of the roots = 16/3  = a + b         square both sides

a^2 + 2ab + b^2  = 256 / 9

a^2 + (34/3) + b^2  =  256 /  9

a^2 + b^2  =  256 / 9  - 34 /3  =  154 / 9

Quadratic  is

x^2  - (154/9)x + 289 / 9

8 * 45  =  18 * x

360  = 18 * x

360  / 18 =  x

20 (min)   = x

Call the side, s

s^2 = 10s + 38

s^2  - 10s  =  38

s^2 -10s + 25 =  38 + 25

(s - 5)^2  = 63       take  both roots

s - 5  = sqrt (63)                   s - 5 =  -sqrt (63)

s = 5 + sqrt (63)                  s =  -5 -sqrt (63)    reject

s ≈ 12.94