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The equation to the side BC of the triangle ABC is x + 5 = 0. If (-3,2) is the orthocentre and origin is circumcentre , then radius is ?? A √51B √ 13C √53

The equation to the side BC of the triangle ABC is x + 5 = 0. If (-3,2) is the orthocentre and origin is circumcentre , then radius is ?? A √51B √ 13C √53

Grade:12

1 Answers

Arun
25757 Points
4 years ago
Dear student
 
When you ask for the radius of the circle you mean the circumcircle? Is that right? 

The orthocentre, centroid, and circumcentre are collinear, all lying on the Euler line. The centroid divides the line segment from orthocentre to circumcentre in ratio 2:1. Dilate the orthocentre with respect to the circumcentre by scale factor 1/3. The image is the centroid. 

centroid(-1, 2/3) 

Line BC was given (though not actually B or C). Call this line s. The centroid divides all medians in ratio 2:1, being twice as far from each vertex as it is from the opposite midpoint. Dilate line s with respect to the centroid by factor -2. This is its image. 

s': x = 7 

Vertex A must lie on this line. The altitude from A must run through the orthocentre, perpendicular to line s. Point A must also lie on this line, labeled k below. 

k (altitude from A): y = 2 

That gives us A(7, 2). 

The circumradius is the distance from the circumcentre to A. 

circumradius = √(7² + 2²) = √(53)

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