how to draw this triangle ?

Question

Here is a problem i am trying to solve ,

Problem :
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triangle ABC is a right angled isosceles triangle and AB is the diameter of a circle . If AC=6 cm , find the area common to the triangle and the circle.


My Question :
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how do i draw the figure for this problem ?

i see AB is the diameter of the circle but I am stuck, where does this point C lies ? does the point C lie on the circle ?
or the point C lies outside of this circle ? I am bit confused to draw the triangle.

and also which two sides will be of same length (isocles) ?

could you please help me to understand this problem and draw the figure ?


please let me know how to draw this figure IN STEPS


Thanks

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Hi, ciric ....as you told C does not lie on the circle , so where does it lie ?

does it lie inside or outside the circle ?

Maria,
Thanks maria for the effort ...but i am not asking the procedure to draw.....i am asking where to put C.

Hope you guys understand my question now.

as the question does not talk about point C , so i am bit confused to draw the triangle

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sofarsogo

you told...."....A right isosceles triangle is a 45-45-90 triangle, the hypotenuse AB is the diameter, so of course C lies on the circle........"

but that could be other way around ! ...nobody is stopping me to put C outside and draw a right-angle triangle !

especially, when the question does not state clearly...how do i assume it ?


well, ok...let me tell you , the answer from my book .........they have assumed C outside of the circle ..........but i am really disatisfied with this figure ............you know mr sofarsog.... , i did exactly the way you thought ......unfortunately this does not match up with the book and hence the answer varies .

do you think , there could be no unique figure ?

Most important thing is to understand the clue given in the problem to come to the exact figure......though i dont see any clue here which could push point C to the ouitside of the circle ....and hence i am dubiuos

Answer 1

ok!
i know how to draw that, but my english are owful, so let s help eatcother:)
i don t know what is the name of that gajet that you use in order to draw a circle...lets name it ''diaviti''!

1. take ''diaviti'' and draw a cicle with 6cm diameter!
2. draw a diameter on your cicle, and name that AB.
3. put ''diaviti'' on the point A and draw a new cicle (6cm dimeter)
4. put ''diaviti'' on the point A and draw another cicle (6cm dimeter)
5. the two last cicles have 2 common points!
6. chose one of those points and name it C.
7. conect the tree points A, B, C, and........
8. you get it!!!!!

have a nice day!

Answer 2

Am I missing something? A right isosceles triangle is a 45-45-90 triangle, the hypotenuse AB is the diameter, so of course C lies on the circle. Remember, an angle whose vertex lies on a circle has measure of 1/2 the intersected arc. If AB is a diameter then arc AB = 180, so if C is drawn on the circle then angle C is 180/2 = 90.

Or, you can see it this way: take the diameter AB and draw the perpendicular from the midpoint (the center of the circle). This intersects the sircle at point P. This is also a radius, so if the center is x then xA = xB = xP = radius = 6/2 = 3. Draw PA and PB, this gives triangles xPA and xPB. These are each right isosceles triangles, so their small angles are 45 degrees. Angle APB is 2 of these, 45 + 45 = 90, so my P is your C.

*EDIT*
OK, without a picture I was assuming AB was the hypotenuse. If it is not then it is a leg. In this case C has to be outside the circle. Say BC is the hypotenuse. Then angle A is 90, so AC is tangent to the circle - it never enters the circle, so C cannot be inside.

Now if AC = 6 I have the same problem as earlier - AC could be the hypotenuse or a leg. My guess is it is a leg, so the diameter of the circle is also 6. If so, the problem is not hard. Draw the radius from the center of the circle parallel to AC. This divides the common area into 2 sections: a sector of 90 degrees and an isosceles right triangle with legs the radius. Radius = 3, so the sector is 9pi/4 and the right triangle is 3*3/2 = 9/2, total area is 9pi/4 + 9/2

If AC is the hypotenuse everything is the same except that 6 is the hypotenuse, so the legs (and the diameter) are 3sqrt(2) instead of 6.

Answer 3

You must remember that when you say that tringle ABC is a right angle triangle that means the angle ABC is right angle. As it is an isoscale it means leangth AB=BC. The AC is the hypotaneous.
Naturally the point C lies out of the circle.

Now how to draw the figure?
Draw the line AC which is of 6 cm.
Then put your protractor on point A and mark 45 degree angle and draw a long line making angle 45 degree at A.
Again draw a 45 degree angle at point C with the protractor.
Let us call the two new lines as AD and CE.
the point at which these two lines will intersect is the point B. And the angle ABC is a right angle.
The length of line AB and BC will be equal and will be 3 times square root of 2. (I couldnot draw the symbol of square root as it is not represented on my keyboard ) . now draw the circle after bisecting the line AB. Name the point of bisection as F. AF is the radius of the circle and the point F is the midpoint of line AB
Once you have drawn the circle you will see that C is out of the circle. Finding the area is easy. Just use the sector and triangle thus formed
It seems all the confusion has arisen due to the uncertanity about deciding the right angle.
Actually there is no confusion at all.
In mathematics it the dictum to name the right angle triangle by keeping the right angle in centre.
so likewise in a right angle triangle ABC, the angle ABC is right angle. Similarly in a right angle triangle DEF the angle DEF is right angle and line DF is hypotaneous.

Answer 4

Hi. Let B be the circle's center. Then the common area will simply be the area of the triangle. 36/2 = 16 sq cm

Edit: Sorry, read AB as radius. Point C does not lie on the circle, only points A and B. The common area is then half the area of the circle.

Answer 5

first u calculate the radius which is pie r squared times 180 degrees then you draw an acute angle to the angle bisector which is parallel to the 90 degree right angle which makes an obtuse angle perpendicular to the straight line.

Answer 6

Maria is right. "diaviti" (dividers) are known as a drawing compass in English.