40 Questions
What is the point of intersection of the medians of a triangle?
Centroid
What is the ratio in which the centroid divides each median?
2:1
What is the line segment that joins the mid-point of a side to the opposite vertex called?
Median
What is the name of the point where the three medians of a triangle intersect?
Centroid
What is the purpose of a median in a triangle?
To join the mid-point of a side to the opposite vertex
What is the property of a centroid with respect to a triangle?
It is the point of intersection of the medians
What is the ratio in which the centroid divides the median in a triangle?
2:1
What is the term used to describe the point where the three medians of a triangle intersect?
Centroid
What is the formula for the area of an equilateral triangle with altitude P?
P^2 / 3
If the side of an equilateral triangle is 'a', what is its area?
a^2 * sqrt(3) / 4
What is the formula for the area of a quadrilateral with given vertices?
1/2 * (x1y2 - x2y1 + x2y3 - x3y2 + x3y4 - x4y3 + x4y1 - x1y4)
How can the area of a polygon be obtained?
By dividing the polygon into triangles and adding their areas
What is the formula for the area of a triangle formed by three points A(x1, y1), B(x2, y2), and C(x3, y3)?
1/2 * |x1*(y2 - y3) + x2*(y3 - y1) + x3*(y1 - y2)|
If the area of a triangle is 2 sq units, what is the value of the expression inside the modulus?
-4
What is the value of x1y2 - x2y3 + x3y1 - x2y1 + x3y2 - x1y3 in the example given?
-4
What is the value of x1y2 - x2y3 + x3*y1 in the example given?
55
What are the coordinates of the third vertex found in the given system of equations?
(10, -2)
What is the point of intersection of internal bisectors of the angles of a triangle called?
Incenter
What is the center of the circle touching all the sides of a triangle called?
Incenter
What is the formula for the coordinates of the incenter of a triangle ABC?
$(ax_1 + bx_2 + cx_3)/(a + b + c), (ay_1 + by_2 + cy_3)/(a + b + c)$
What is the ratio in which the angle bisector divides the opposite sides in a triangle?
Ratio of remaining sides
What is the ratio in which the incenter divides the angle bisectors in a triangle?
(b + c) : a, (c + a) : b, (a + b) : c
What are the points A, B, and C in the diagram?
Vertices of the triangle
What is the circle touching all the sides of a triangle called?
Incircle
What is the area of the triangle ACD?
17.5 sq.units
What is the area of the quadrilateral ABCD?
28 sq.units
What is the condition for the vertices of an equilateral triangle?
All vertices cannot be rational
What is the formula used to calculate the area of triangle ACD?
1/2 |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|
What is the value of the expression |(–4) (–2 – 3) + 3 ( 3 + 2) + 2 (–2 + 2)|?
35
What are the coordinates of the vertices of the triangle ABC?
A(x1, y1), B(x2, y2), C(x3, y3)
What is the purpose of the given example?
To prove that the coordinates of the vertices of an equilateral triangle cannot all be rational
What is the formula used to calculate the area of a quadrilateral?
ar (ΔABC) + ar (ΔACD)
What are the coordinates of the circumcenter of a triangle formed by three points?
($x_1 sin^2A + x_2 sin^2B + x_3 sin^2C$, $y_1 sin^2A + y_2 sin^2B + y_3 sin^2C$) / (sin^2A + sin^2B + sin^2C)
What is the condition for the points A, B, and C to be collinear?
AP^2 = BP^2 = CP^2
What is the equation of the line formed by the points A and B?
4x - 2y + 5 = 0
What is the value of x in the circumcenter of the triangle formed by the points (0, 0), (-2, 1), and (-3, 2)?
3/11
What is the equation of the line formed by the points A and C?
6x - 4y + 13 = 0
What is the value of y in the circumcenter of the triangle formed by the points (0, 0), (-2, 1), and (-3, 2)?
11/3
What is the circumcenter of the triangle formed by the points A, B, and C?
The point where the three perpendicular bisectors of the sides of the triangle intersect
What is the relation between the distances AP, BP, and CP?
AP = BP = CP
Study Notes
Area of Equilateral Triangle
- If the altitude of an equilateral triangle is P, then its area is P²/3 square units.
- If 'a' is the side of an equilateral triangle, then its area is a²√3/4 square units.
Area of Quadrilateral
- If the vertices of a quadrilateral are A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4), then its area is (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) + x4(y2 - y1) + x1(y4 - y2)|.
- The area of a polygon can be obtained by dividing the polygon into disjoint triangles and then adding their areas.
Coordinate Geometry
- To find the area of a triangle formed by three points, use the formula (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.
- Examples: Find the area of the triangle formed by the points A(5, 2), B(4, 7), and C(7, -4). Find the area of quadrilateral ABCD.
Properties of Triangles
- The coordinates of the vertices of an equilateral triangle cannot all be rational.
- The centroid of a triangle is the point of intersection of the medians, and it divides each median in the ratio of 2:1.
- The incenter of a triangle is the point of intersection of the internal bisectors of the angles, and it is the center of the circle touching all the sides of the triangle.
- The coordinates of the incenter I are (ax + bx2 + cx3 / a + b + c, ay1 + by2 + cy3 / a + b + c), where a, b, and c are the sides of the triangle.
Circumcenter
- The coordinates of the circumcenter 'O' are (x1 sin²A + x2 sin²B + x3 sin²C / sin²A + sin²B + sin²C, y1 sin²A + y2 sin²B + y3 sin²C / sin²A + sin²B + sin²C).
- The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle.
- Examples: Find the coordinates of the circumcenter of the triangle formed by the points (0, 0), (-2, 1), and (-3, 2).
This quiz is about the area of an equilateral triangle. It covers the formulas for calculating the area of an equilateral triangle using its altitude and side length.
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