A triangle has corners at (4 ,7 )(4,7), (3 ,4 )(3,4), and (8 ,9 )(8,9). What is the area of the triangle's circumscribed circle?

2 Answers
Narad T.
Jun 7, 2017

The area of the circumscribed circle is =78.54=78.54

Explanation:

To calculate the area of the circle, we must calculate the radius rr of the circle

Let the center of the circle be O=(a,b)O=(a,b)

Then,

(4-a)^2+(7-b)^2=r^2(4−a)2+(7−b)2=r2.......(1)(1)

(3-a)^2+(4-b)^2=r^2(3−a)2+(4−b)2=r2..........(2)(2)

(8-a)^2+(9-b)^2=r^2(8−a)2+(9−b)2=r2.........(3)(3)

We have 33 equations with 33 unknowns

From (1)(1) and (2)(2), we get

16-8a+a^2+49-14b+b^2=9-6a+a^2+16-8b+b^216−8a+a2+49−14b+b2=9−6a+a2+16−8b+b2

2a+6b=402a+6b=40

a+3b=20a+3b=20.............(4)(4)

From (2)(2) and (3)(3), we get

9-6a+a^2+16-8b+b^2=64-16a+a^2+81-18b+b^29−6a+a2+16−8b+b2=64−16a+a2+81−18b+b2

10a+10b=12010a+10b=120

a+b=12a+b=12..............(5)(5)

From equations (4)(4) and (5)(5), we get

12-b+3b=2012−b+3b=20

2b=82b=8

b=8/2=4b=82=4

a=12-b=12-4=8a=12−b=12−4=8

The center of the circle is =(8,4)=(8,4)

r^2=(4-a)^2+(7-b)^2=(4-8)^2+(7-4)^2r2=(4−a)2+(7−b)2=(4−8)2+(7−4)2

=16+9=16+9

=25=25

The area of the circle is

A=pi*r^2=25*pi=78.54A=π⋅r2=25⋅π=78.54

Dean R.
Jun 29, 2018

(4-3)^2+(7-4)^2=10(4−3)2+(7−4)2=10

(8-3)^2+(9-4)^2=50(8−3)2+(9−4)2=50

(8-4)^2+(9-7)^2=20(8−4)2+(9−7)2=20

pi r^2 = {pi (10)(50)(20) }/{ 4(10)(50) - (20-10-50)^2} = 25 pi πr2=π(10)(50)(20)4(10)(50)−(20−10−50)2=25π

Explanation:

Here's the shortcut.

The circumcircle is just the circle through the three vertices; the triangle almost doesn't matter. Except, miraculously, the circumradius rr equals the product of the triangle sides a,b,ca,b,c divided by four times the triangle's area AA.

r = {abc}/{4A}r=abc4A

It's much more useful squared, and we're looking for pi r^2πr2 anyway.

pi r^2 = {pi a^2 b^2 c^2}/{16A^2}πr2=πa2b2c216A2

The coordinates give the squared distances easily. Archimedes' Theorem relates the squared distances to the triangle area:

16A^2 = 4a^2b^2 - (c^2-a^2-b^2)^216A2=4a2b2−(c2−a2−b2)2

So,

pi r^2 = {pi a^2 b^2 c^2}/{ 4a^2b^2 - (c^2-a^2-b^2)^2}πr2=πa2b2c24a2b2−(c2−a2−b2)2

We form the squared distances from pairs of points (4,7),(3,4), ( 8,9) (4,7),(3,4),(8,9)

a^2=(4-3)^2+(7-4)^2=10a2=(4−3)2+(7−4)2=10

b^2=(8-3)^2+(9-4)^2=50b2=(8−3)2+(9−4)2=50

c^2=(8-4)^2+(9-7)^2=20c2=(8−4)2+(9−7)2=20

pi r^2 = {pi (10)(50)(20) }/{ 4(10)(50) - (20-10-50)^2} = 25 pi πr2=π(10)(50)(20)4(10)(50)−(20−10−50)2=25π

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