30 Questions
What is the value of r in the polar coordinates of the point (3, 1)?
2
In any triangle ABC, what is the condition for AB + BC > AC to be true?
always true
What is the condition for a triangle ABC to be equilateral?
AB = BC = CA
What is the formula for the area of a triangle in terms of its sides?
a² + b² - 2ab cos(θ)
What is the condition for a quadrilateral ABCD to be a parallelogram?
AB = DC and AD = BC
What is the point where the x-axis and y-axis intersect called?
Origin of coordinates
What are the coordinates of the origin in a rectangular Cartesian coordinate system?
(0, 0)
What is the coordinate of the orthocentre of a triangle with vertices (0, 3), (–3, 0) and (3, 0)?
(0, 3)
If the coordinates of a point P are (x, y), what is the value of x if the point lies on the y-axis?
x = 0
If the centroid and circumcentre of a triangle are (3, 3) and (6, 2), what is the coordinate of the orthocentre?
(–3, 5)
What is the formula to find the distance between two points P(x1, y1) and Q(x2, y2) in a rectangular Cartesian coordinate system?
PQ = √((x2 - x1)² + (y2 - y1)²)
What is the formula for the area of a triangle with vertices A(x1, y1), B(x2, y2) and C(x3, y3)?
(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2
If the polar coordinates of a point P are (r, θ), what is the value of y in terms of r and θ?
y = r sin θ
What is the area of an equilateral triangle with side length 'a'?
(a^2 * sqrt(3)) / 4
What is the formula for the area of a triangle with base 'b' and perpendicular height 'p'?
(b * p) / 2
What is the condition for the points (0, 8/3), (1, 3), and (82, 30) to be collinear?
cos 2θ ≠ 1
What is the effect of shifting the origin to a point (α, β) without rotation of axes?
The new equation of the curve is obtained by replacing x with x - α and y with y - β
What is the result of rotating the axes without changing the origin?
The coordinates of the points remain unchanged
What is the relationship between the vertices A, B, and C if AB + BC = AC?
The points A, B, and C are collinear
What is the effect of shifting the origin on the equation of a curve?
The equation of the curve is transformed by replacing x with x + α and y with y + β
What is the condition for the points A, B, and C to form an equilateral triangle?
AB = BC = CA
What is the result of rotating the axes by 90 degrees without changing the origin?
The x-axis and y-axis are interchanged
What is the relationship between the points A, B, and C if they are collinear?
AB + BC = AC
What is the effect of shifting the origin on the coordinates of a point?
The coordinates of the point are translated by (α, β)
What is the property of the points A, B, and C if they form a right-angled triangle?
AB^2 + BC^2 = CA^2
If the centroid of a triangle ABC divides its median in the ratio k : 1, then what is the value of k?
2
A line divides two points (2, -3) and (-5, 6) in the ratio 2:3 internally. What is the coordinate of the point of division?
(1, 2)
If the vertices of a triangle are A(x1, y1), B(x2, y2), and C(x3, y3), then what are the coordinates of its centroid?
((x1+x2+x3)/3, (y1+y2+y3)/3)
A line divides two points (1, 1) and (5, 7) in the ratio 3:4 internally. What is the equation of the line?
x + y = 4
If the ratio in which a line divides two points (x1, y1) and (x2, y2) is k:1, then what is the coordinate of the point of division?
((kx2 + x1)/(k + 1), (ky2 + y1)/(k + 1))
Practice finding the orthocentre of a triangle given its vertices and understanding the relationship between the centroid, circumcentre, and orthocentre. Enhance your problem-solving skills in coordinate geometry.
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