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Math Questions (geometry)

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5 Also this: Next, And finally, Jun 29, 2018
edited by AWESOMEEE  Jun 29, 2018

#1
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Here's the second one

MIR  is an inscribed angle

So....since it subtends minor arc  MR, the  angular measure of this arc  is twice 18° = 36°

So......the length  of MR  is given by :

2 pi  *  radius *  (36 / 360)  =

2pi * 80  * (1  /10) =

(160 / 10) pi  =

16 pi   cm   Jun 30, 2018
edited by CPhill  Jun 30, 2018
edited by CPhill  Jun 30, 2018
#2
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Doesn't seem to work. I entered in 3.6pi, then 18pi/5.

I can't enter anymore, but I am still curious for the solution.

AWESOMEEE  Jun 30, 2018
edited by AWESOMEEE  Jun 30, 2018
#3
+1   CPhill  Jun 30, 2018
#4
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Here's the first one....there might be an easier way to do it, however  !!!

The area of  ABC  is   (1/2) (16)^2 (√3/2)  =  64√3  cm

So...the area  of  equilateral triangle   PQR is   (1/4)  of this  = 16√3 cm^2

And we can find the   length of  PR  by using the area formula for PQR

(1/2) (PR)^2 (√3 /2 )  =  16√3

(1/2) PR^2 (1/2)  = 16

PR^2  / 4  = 16

PR^2  = 64

PR  = 8  cm

And note that we have trapezoid BPRC  which has the same area as the small equilateral triangle

So   area  of trapezoid  BPRC  = area of triangle PQR

also

(1/2) (height) ( sum of the bases)  = 16√3

height ( 16 + 8)  = 32√3

height (24)  = 32√3

height  =  32√3 / 24   =   (4/3)√3  cm

So...note  that if we draw a perpendicular  from P  to BC  this  will be the height of the trapezoid and will  also form one leg of a right triangle  with BP  as the hypotenuse

And the other leg length will be  ( BC  - PR)  / 2   = (16 - 8) /2  =  8/2  = 4

So....using the Pythagorean Theorem...we can find  BP  as

BP  =   √ [ 4^2  +  [ 4/3 * √3 ]^2  ]  =  √ [ 16  +  16/3  ]  =   4 √ [ 1 + 1/3] =

4 √ [ 4/3 ]  = 8√ [ 1/3 ]  =  8 / √3  cm   =  8√3 / 3  cm   Jun 30, 2018
#5
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Here's the third one

Orient the paper so that 22 is the width and 39 is the height

Note that  the number of retangles of (2 * 3)  that can  be cut from this sheet is

(39/3)  * ( 22/2)  =

13  * 11  =

143

But each of the rectangles  can  be divided into two triangles with bases of 2 and heights of 3

So....the total number of triangles  is  143 * 2  =  286   Jun 30, 2018