What is the value of a if the area of a circle formed by the points (7, 0), (11, 0) and (0, a) is 36 squared cm.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The area of a triangle is given by (1/2)*base*height. In the given question the vertices of the triangle are at (7, 0), (11, 0) and (0, a). The base of the triangle is equal to the distance between the points (7, 0) and (11, 0). As the y coordinate is equal to 0 we can find the distance between the points as 11-7 = 4.

Area = (1/2)*base*height

=> height = 2*area/base

The height of the triangle is 2*area/base = 2*36/4 = 18.

Therefore the point (0, a) is (0, 18)

The required value of a is 18.

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neela | High School Teacher | (Level 3) Valedictorian

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Since  the points (7,0) and(11,0) are the points on the circle,

Let (h,k) be the center . Then  the centre (h,k) is equidistant from (7,0) and (11,0).

So,  (h-7)^2 +(k-0)^2 = (h-11)^2+k(0)^2 , k^2 gets cancelled on both sides.

h^2-14h+49 = h^2-22h+121,  h^2 gets cancelled on both sides.

-14h+49 = -22h+121

22h-14h = 121-49 = 72.

8h = 72, or h = 72/8 = 9.

Also are of the circle is 36 sq cm.

Therefore pir^2 = 36. So r^2 = 36/pi.

Now since  (7,0) is on the circle,

(7-h)^2 +(0-k)^2 = r^2.

Therefore  k^2 = r^2-(7-h)^2 = r^2 -(7-9)^2 = r^2-4.

So k^2 = (36/pi - 4).

Therefore k = sqrt(3/pi-4) , Or k = -sqrt(36/pi -4).

To find the value of a if (0, a) is a point on the circle.

Now the centre of the circle is (h,k) = (9 , sqrt(36/pi -4) ) or  (9 , -sqrt(36/pi -4) ).

(0,a) is point on the circle with centre ( 9, sqrt(36/pi -4) )

 (0-9)^2 + (a-sqrt(36/pi - 4) ^2  = 36/pi

81 + a^2 -2asqrt(36/-4) + 36/pi -4 = 36/pi

a^2 -2a sqrt(36/pi-4) -4 = 0

 a1 = {2+ sqrt { 36/pi-4 +16)}/2 = {1+sqrt(36/pi +12)}

 a2 = {1-sqrt(36/pi +12).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The enunciation is ambiguous. Since the value of area is specified, it is not necessary to give the coordinates of the other points.

What is also ambiguous is the fact that the given value of the area of the circle is not depending on pi = 3.14.

The area of the circle is 36 squared cm.

We know that the area of the circle is:

A = pi*r^2, where r is the radius of the circle.

36 = pi*r^2

We'll apply square root both sides:

6 = r sqrt pi

r = 6/sqrt pi

r = 3.38 cm

Since the point is on the circle, we'll write the equation of the circle:

x^2 + (y-a)^2 = 11.42

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