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What are the coordinates of the centroid of triangle ABC with vertices (x1, y1), (x2, y2), and (x3, y3)?
If the mid-point D of segment BC is given by the coordinates ($rac{x2 + x3}{2}, rac{y2 + y3}{2}$), what is the ratio that the centroid G divides AD?
What are the coordinates of point G, the centroid of triangle ABC, expressed in terms of x1, x2, and x3?
When considering the point D that bisects angle BAC, what geometric position does it represent?
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Given the coordinates of points A, B, and C of a triangle, how can the coordinates of the centroid G be calculated?
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What is the relationship of the centroid to the orthocentre and circumcentre in terms of distance?
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In which type of triangle do the centroid, incentre, orthocentre, and circumcentre coincide?
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What does the nine-point circle touch?
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What is true about the relationship between the orthocentre, centroid, and circumcentre?
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In an isosceles triangle, which points lie on the same line?
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How does the orthocentre relate to the right angle in a right-angled triangle?
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What does HA represent in relation to the orthocentre?
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What is the formula for the distance of the orthocentre from vertex A?
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What is the definition of inclination in relation to a line intersecting the x-axis?
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What is the slope of a line if its inclination angle θ is defined as π/2?
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Which of the following correctly defines the slope of a non-vertical line?
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If two points on a line are (x1, y1) and (x2, y2), how is the slope calculated?
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For a line with a positive slope, which of the following statements is true regarding angle θ?
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What is the slope of the x-axis?
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Which of the following describes the relationship when three points are collinear?
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Given the slope formula m = tan(θ), what is the slope when θ is 0 degrees?
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What is the image of the point P(x1, y1) with respect to the x-axis?
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Given the equation of the line ax + by + c = 0, which condition indicates that the point P(x1, y1) is on the same side of the line as the origin?
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What is the formula used to find the image of point A(x1, y1) with respect to the line ax + by + c = 0?
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What is the correct equation of the line that is at a distance of 3 from the origin and makes an angle of 30° with the positive x-axis?
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If point P(x1, y1) is at (2, 3), what is its image with respect to the y-axis?
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What is the image of the point P(x, y) with respect to the line y = x?
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What distance does the point R1 divide the line joining P(x1, y1) and Q(x2, y2) in the ratio m:n represent?
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What does the formula $x ext{cos} heta + y ext{sin} heta = P$ determine in relation to lines?
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What is the equation of the bisector of the angle containing the origin if the lines are given by the equations with positive constant terms?
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Given two lines intersecting with positive constants, when is the angle between them acute?
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What determines if the angle bisector of two lines contains the origin or not?
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For two lines defined by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, how do you determine the acute angle bisector?
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When comparing the acute angle bisector with the obtuse angle bisector, what condition indicates that the angle is obtuse?
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What is the correct form of the bisector equation for the obtuse angle between two lines?
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What adjustment is made to the equations of the lines to check if the bisectors contain the origin?
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Description
This quiz covers the major triangle centers including the centroid, incentre, and orthocentre. Learn how to calculate these points based on the triangles' vertices and understand their significance within triangle geometry. Test your knowledge on the properties and formulas associated with these important points.