Question

1. in an obtuse - triangle, where is the circumcenter located? a. inside the triangle b. outside the triangle c. at the centroid d. at the incenter 27. which point lies on the circle represented by the equation (x - 1)^2+(y + 2)^2 = 25? a. (3,1) b. (0,0) c. (6, - 2) d. (0, - 7)

Explanation:

Step1: Recall circum - center property

The circum - center of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. In an obtuse - angled triangle, the circum - center lies outside the triangle.

Step2: Analyze circle equation

The equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center of the circle and \(r\) is the radius. For the circle \((x - 1)^2+(y + 2)^2=25\), the center is \((1,-2)\) and radius \(r = 5\).

Step3: Test points in circle equation

For point \((3,1)\): \((3 - 1)^2+(1+ 2)^2=4 + 9=13
eq25\).
For point \((0,0)\): \((0 - 1)^2+(0 + 2)^2=1 + 4=5
eq25\).
For point \((6,-2)\): \((6 - 1)^2+(-2 + 2)^2=25+0 = 25\).
For point \((0,-7)\): \((0 - 1)^2+(-7 + 2)^2=1 + 25=26
eq25\).

Answer:

1. B. Outside the triangle
2. C. \((6,-2)\)