Question 1199419
.


How you formulate the problem in your post,  it makes me cry: 
so mathematically illiterate the formulation is.


Therefore, I will edit it to present it in a decent shape.


<pre>
    In the diagram, ABC is a right triangle with the right angle at B.
    Angle C is 30°. The leg BC is 5 cm long. 
    Circular arc DE with the center at B is tangent to hypotenuse AC. 
    Find the area, in cm2, of the quarter circle BDE.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Next, the solution by @josgarithmetic is incorrect,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;starting from the second line of his post. 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;AB is not 10 cm, as he mistakenly states in his post.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, ignore his solution, &nbsp;since it is wrong.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I came to bring you a correct solution.



<pre>
Since the angle A is 30°, the hypotenuse AC is 10 cm and the leg AB is  10*cos(30°) = {{{10*(sqrt(3)/2)}}} = {{{(5*sqrt(3))}}}.

Then the area of the triangle ABC is  {{{(1/2)*abs(BC)*abs(AC)}}} = {{{(1/2)*(5*(5*sqrt(3))))}}} = {{{(25*sqrt(3))/2}}}.


Next, the radius of the circle "r" is the height of the triangle ABC, drawn from vertex B to the hypotenuse AC.


Therefore, the area of the triangle ABC in other form can be presented as  {{{(1/2)*r*10}}}  cm^2.


The area is the same, so we can write this equation

    {{{(1/2)*r*10}}} = {{{(25*sqrt(3))/2}}}  cm^2.


It gives

          r    = {{{(5*(sqrt(3))/2)}}}  cm.


Now the area of the quarter of the circle is

    {{{(pi*r^2)/4}}} = {{{(pi*25*3)/(4*4))}}} = {{{(3.14*75)/16}}} = 14.72  cm^2  (rounded).    <U>ANSWER</U>
</pre>

Solved.


---------------------


No one of the listed numbers in your post is not even close to the right answer.



It is easy to see, without making long calculations, that all listed numbers in your post are not even close to the right answer.


Indeed, the radius of the circle is, obviously, shorter than 5 cm.


Hence, the area of the quarter of the circle &nbsp;MUST &nbsp;be less than &nbsp;&nbsp;{{{(pi*r^2)/4}}} = {{{(3.14*5^2)/4}}} = 19.625 cm^2.


Compare it with your numbers.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Then &nbsp;REPORT &nbsp;to &nbsp;YOUR &nbsp;PROFESSOR &nbsp;that all your 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;attached answers &nbsp;ARE &nbsp;INCORRECT &nbsp;NUMBERS.