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Questions and Answers
What is the value of k for the points (8, 1), (k, -4), and (2, -5) to form an area of 0?
What is the value of k for the points (8, 1), (k, -4), and (2, -5) to form an area of 0?
What are the coordinates of the midpoint D between points (0, -1) and (2, 1)?
What are the coordinates of the midpoint D between points (0, -1) and (2, 1)?
What are the coordinates of the point P, which divides the segment joining (4, -1) and (-2, -3) in the ratio 1:2?
What are the coordinates of the point P, which divides the segment joining (4, -1) and (-2, -3) in the ratio 1:2?
What is the x-coordinate of the point Q, which divides the line segment joining (4, -1) and (-2, -3) in the ratio 2:1?
What is the x-coordinate of the point Q, which divides the line segment joining (4, -1) and (-2, -3) in the ratio 2:1?
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If Niharika posts a green flag at coordinates (2, 25), what fraction of the distance AD has she run?
If Niharika posts a green flag at coordinates (2, 25), what fraction of the distance AD has she run?
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What are the coordinates of Preet's red flag when he has run 1/5 of the distance AD?
What are the coordinates of Preet's red flag when he has run 1/5 of the distance AD?
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How far apart, in meters, are Niharika's green flag and Preet's red flag?
How far apart, in meters, are Niharika's green flag and Preet's red flag?
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Where should Rashmi post her blue flag to be exactly halfway between the other two flags?
Where should Rashmi post her blue flag to be exactly halfway between the other two flags?
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What is the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6)?
What is the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6)?
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What is the y-coordinate of point P, where Rashmi posts her blue flag?
What is the y-coordinate of point P, where Rashmi posts her blue flag?
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What are the coordinates of the midpoint D of side BC in triangle ABC?
What are the coordinates of the midpoint D of side BC in triangle ABC?
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What is the area of triangle ABD?
What is the area of triangle ABD?
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In the equation of the line 2x + y - 4 = 0, what does k represent when the line divides segment AB?
In the equation of the line 2x + y - 4 = 0, what does k represent when the line divides segment AB?
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What is the value of k when the line divides segment AB in the given problem?
What is the value of k when the line divides segment AB in the given problem?
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Which equation represents the condition for the points (x, y), (1, 2), and (7, 0) to be collinear?
Which equation represents the condition for the points (x, y), (1, 2), and (7, 0) to be collinear?
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If areas of triangles ABD and ACD are both 3 square units, what does this indicate about median AD?
If areas of triangles ABD and ACD are both 3 square units, what does this indicate about median AD?
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Which of the following points lies on the vertical line drawn through point B?
Which of the following points lies on the vertical line drawn through point B?
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What is the area of triangle ACD when applying the same formula as for triangle ABD?
What is the area of triangle ACD when applying the same formula as for triangle ABD?
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What is the center of the circle with equations for the radii OA, OB, and OC given the coordinates of point O?
What is the center of the circle with equations for the radii OA, OB, and OC given the coordinates of point O?
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What is the value of y when equating OB and OC, leading to the equation derived from the problem?
What is the value of y when equating OB and OC, leading to the equation derived from the problem?
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Given two opposite vertices of a square at (-1, 2) and (3, 2), what is the length of the diagonal AC?
Given two opposite vertices of a square at (-1, 2) and (3, 2), what is the length of the diagonal AC?
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What is the coordinate of point O, the intersection of diagonals AC and BD, for the given square vertices?
What is the coordinate of point O, the intersection of diagonals AC and BD, for the given square vertices?
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After finding the coordinates of point D, which equation represents the equality of sides AD and CD?
After finding the coordinates of point D, which equation represents the equality of sides AD and CD?
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What must be solved to determine the coordinates of point D in the square?
What must be solved to determine the coordinates of point D in the square?
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What do the diagonals of a square do relative to one another?
What do the diagonals of a square do relative to one another?
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If the x-axis divides the line segment joining points A (1, -5) and B (-4, 5), what are the coordinates of the point of division?
If the x-axis divides the line segment joining points A (1, -5) and B (-4, 5), what are the coordinates of the point of division?
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What are the coordinates of point C if the parallelogram has vertices A(1, 2), B(4, y), D(3, 5) and the midpoint condition is satisfied?
What are the coordinates of point C if the parallelogram has vertices A(1, 2), B(4, y), D(3, 5) and the midpoint condition is satisfied?
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Given the center of a circle at (2, -3) and point B at (1, 4), what are the coordinates of point A if AB is the diameter?
Given the center of a circle at (2, -3) and point B at (1, 4), what are the coordinates of point A if AB is the diameter?
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If point P divides the line segment AB in the ratio 3:4 where A is at (-2, -2) and B is at (2, -4), what is the y-coordinate of point P?
If point P divides the line segment AB in the ratio 3:4 where A is at (-2, -2) and B is at (2, -4), what is the y-coordinate of point P?
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Which of the following pairs of coordinates represents the points that divide the segment joining A(1, -5) and B(-4, 5) by the x-axis?
Which of the following pairs of coordinates represents the points that divide the segment joining A(1, -5) and B(-4, 5) by the x-axis?
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In a parallelogram with vertices A(1, 2), B(4, y), C(x, 6), and D(3, 5), if the midpoint condition holds, what is the value of y?
In a parallelogram with vertices A(1, 2), B(4, y), C(x, 6), and D(3, 5), if the midpoint condition holds, what is the value of y?
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What is the ratio in which the line segment joining points (-2, -2) and (2, -4) is divided by point P if AP is 3/7 AB?
What is the ratio in which the line segment joining points (-2, -2) and (2, -4) is divided by point P if AP is 3/7 AB?
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What are the coordinates of point D which divides line segment AB in the ratio 1:3?
What are the coordinates of point D which divides line segment AB in the ratio 1:3?
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What is the area of triangle ABC given the vertices A (4, 6), B (1, 5), and C (7, 2)?
What is the area of triangle ABC given the vertices A (4, 6), B (1, 5), and C (7, 2)?
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What is the ratio of the area of triangle ADE to the area of triangle ABC?
What is the ratio of the area of triangle ADE to the area of triangle ABC?
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If the vertices of triangle ABC are A(4, 2), B(6, 5), C(1, 4), what are the coordinates of point D, the midpoint of line segment BC?
If the vertices of triangle ABC are A(4, 2), B(6, 5), C(1, 4), what are the coordinates of point D, the midpoint of line segment BC?
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What are the coordinates of point P on AD such that AP:PD = 2:1?
What are the coordinates of point P on AD such that AP:PD = 2:1?
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If the coordinates of points Q and R divide medians BE and CF in the ratio 2:1, which of the following best describes their positions?
If the coordinates of points Q and R divide medians BE and CF in the ratio 2:1, which of the following best describes their positions?
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What is the centroid of triangle ABC with vertices A (x1, y1), B (x2, y2), and C (x3, y3)?
What is the centroid of triangle ABC with vertices A (x1, y1), B (x2, y2), and C (x3, y3)?
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Which of the following accurately reflects the properties of medians in triangle geometry?
Which of the following accurately reflects the properties of medians in triangle geometry?
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What is the area of the triangle formed by points (0, -1), (2, 1), and (0, 3)?
What is the area of the triangle formed by points (0, -1), (2, 1), and (0, 3)?
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What is the area of ΔDEF, which is formed by joining the midpoints of ΔABC?
What is the area of ΔDEF, which is formed by joining the midpoints of ΔABC?
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What is the ratio of the area of the triangle formed by the midpoints to the area of the original triangle?
What is the ratio of the area of the triangle formed by the midpoints to the area of the original triangle?
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What is the total area of the quadrilateral with vertices (-4, -2), (-3, -5), (3, -2), and (2, 3)?
What is the total area of the quadrilateral with vertices (-4, -2), (-3, -5), (3, -2), and (2, 3)?
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When the median of a triangle divides it into two triangles, what can be said about their areas?
When the median of a triangle divides it into two triangles, what can be said about their areas?
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What is the calculated area for triangle ΔABC with vertices A (4, -6), B (3, -2), and C (5, 2)?
What is the calculated area for triangle ΔABC with vertices A (4, -6), B (3, -2), and C (5, 2)?
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What expression is used to find the area of a triangle given its vertices?
What expression is used to find the area of a triangle given its vertices?
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What are the coordinates of the midpoints D, E, and F for the triangle with vertices A(0, -1), B(2, 1), and C(0, 3)?
What are the coordinates of the midpoints D, E, and F for the triangle with vertices A(0, -1), B(2, 1), and C(0, 3)?
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Study Notes
Coordinate Geometry Exercises
- Distance Formula: Used to find the distance between two points (x₁, y₁) and (x₂, y₂). The formula is √((x₂ - x₁)² + (y₂ - y₁)²).
- Midpoint Formula: Finds the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂). Formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Section Formula: Calculates the coordinates of a point dividing a line segment in a given ratio. If a point divides a line segment in the ratio m:n, the coordinates are ((nx₁ + mx₂)/(m+n), (ny₁ + my₂)/(m+n)).
- Collinear Points: Points are collinear when they lie on the same straight line. The area of the triangle formed by these points is zero.
Isosceles Triangle Check
- Identifying Isosceles: To determine if three points form an isosceles triangle, calculate the distance between each pair of points. If any two distances are equal, the triangle is isosceles.
Quadrilaterals
- Identifying Quadrilaterals: Analyze the distances between the vertices of a quadrilateral. Equal sides suggest a rhombus, equal opposite sides imply a parallelogram. Calculation of lengths of the diagonals helps distinguish shapes. Lengths of opposite sides indicate parallelograms. Diagonals equalling or not equalling each other provide information about different shapes (squares, rectangles).
Area of Triangles and Quadrilaterals
- Triangle Area: To find the area of a triangle, use the formula Area= 1/2|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| , where (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of the vertices.
- Quadrilateral Area: Divide a quadrilateral into triangles for calculating its area. Alternatively use the formula Area= 1/2|x₁(y₂-y₄)+x₂(y₃-y₁)+x₃(y₄-y₂)+x₄(y₁-y₃)| . The area of a rhombus is 1/2 * (product of diagonals).
Problems Involving X-axis
- Point on the X-axis: Any point on the x-axis has a y-coordinate of 0.
- Distance: The x-coordinate of a point on the x-axis is equidistant from given coordinate points (x, y) is given by a specific formula
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Description
Test your knowledge on various coordinate geometry concepts including calculating distances, midpoints, and identifying shapes such as isosceles triangles and quadrilaterals. This quiz will challenge your understanding of the foundational formulas used in geometry.
