Equilateral Triangle Area

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Questions and Answers

What is the point of intersection of the medians of a triangle?

Incenter
Orthocenter
Circumcenter
Centroid (correct)

What is the ratio in which the centroid divides each median?

1:1
3:2
3:1
2:1 (correct)

What is the line segment that joins the mid-point of a side to the opposite vertex called?

Median of a hypotenuse
Altitude
Median (correct)
Perpendicular bisector

What is the name of the point where the three medians of a triangle intersect?

Centroid (D) Signup and view all the answers

What is the purpose of a median in a triangle?

To join the mid-point of a side to the opposite vertex (C) Signup and view all the answers

What is the property of a centroid with respect to a triangle?

It is the point of intersection of the medians (A) Signup and view all the answers

What is the ratio in which the centroid divides the median in a triangle?

2:1 (B) Signup and view all the answers

What is the term used to describe the point where the three medians of a triangle intersect?

Centroid (C) Signup and view all the answers

What is the formula for the area of an equilateral triangle with altitude P?

P^2 / 3 (D) Signup and view all the answers

If the side of an equilateral triangle is 'a', what is its area?

a^2 * sqrt(3) / 4 (A) Signup and view all the answers

What is the formula for the area of a quadrilateral with given vertices?

1/2 * (x1y2 - x2y1 + x2y3 - x3y2 + x3y4 - x4y3 + x4y1 - x1y4) (B) Signup and view all the answers

How can the area of a polygon be obtained?

By dividing the polygon into triangles and adding their areas (B) Signup and view all the answers

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)| (B) Signup and view all the answers

If the area of a triangle is 2 sq units, what is the value of the expression inside the modulus?

-4 (B) Signup and view all the answers

What is the value of x1y2 - x2y3 + x3y1 - x2y1 + x3y2 - x1y3 in the example given?

-4 (C) Signup and view all the answers

What is the value of x1y2 - x2y3 + x3*y1 in the example given?

55 (D) Signup and view all the answers

What are the coordinates of the third vertex found in the given system of equations?

(10, -2) (A) Signup and view all the answers

What is the point of intersection of internal bisectors of the angles of a triangle called?

Incenter (A) Signup and view all the answers

What is the center of the circle touching all the sides of a triangle called?

Incenter (D) Signup and view all the answers

What is the formula for the coordinates of the incenter of a triangle ABC?

$(ax_1 + bx_2 + cx_3)/(a + b + c), (ay_1 + by_2 + cy_3)/(a + b + c)$ (A) Signup and view all the answers

What is the ratio in which the angle bisector divides the opposite sides in a triangle?

Ratio of remaining sides (B) Signup and view all the answers

What is the ratio in which the incenter divides the angle bisectors in a triangle?

(b + c) : a, (c + a) : b, (a + b) : c (D) Signup and view all the answers

What are the points A, B, and C in the diagram?

Vertices of the triangle (D) Signup and view all the answers

What is the circle touching all the sides of a triangle called?

Incircle (D) Signup and view all the answers

What is the area of the triangle ACD?

17.5 sq.units (B) Signup and view all the answers

What is the area of the quadrilateral ABCD?

28 sq.units (A) Signup and view all the answers

What is the condition for the vertices of an equilateral triangle?

All vertices cannot be rational (C) Signup and view all the answers

What is the formula used to calculate the area of triangle ACD?

1/2 |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))| (D) Signup and view all the answers

What is the value of the expression |(–4) (–2 – 3) + 3 ( 3 + 2) + 2 (–2 + 2)|?

35 (A) Signup and view all the answers

What are the coordinates of the vertices of the triangle ABC?

A(x1, y1), B(x2, y2), C(x3, y3) (A) Signup and view all the answers

What is the purpose of the given example?

To prove that the coordinates of the vertices of an equilateral triangle cannot all be rational (A) Signup and view all the answers

What is the formula used to calculate the area of a quadrilateral?

ar (ΔABC) + ar (ΔACD) (B) Signup and view all the answers

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) (D) Signup and view all the answers

What is the condition for the points A, B, and C to be collinear?

AP^2 = BP^2 = CP^2 (B) Signup and view all the answers

What is the equation of the line formed by the points A and B?

4x - 2y + 5 = 0 (A) Signup and view all the answers

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 (B) Signup and view all the answers

What is the equation of the line formed by the points A and C?

6x - 4y + 13 = 0 (C) Signup and view all the answers

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 (A) Signup and view all the answers

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 (B) Signup and view all the answers

What is the relation between the distances AP, BP, and CP?

AP = BP = CP (D) Signup and view all the answers

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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).

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