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?

    <p>Centroid</p> Signup and view all the answers

    What is the purpose of a median in a triangle?

    <p>To join the mid-point of a side to the opposite vertex</p> Signup and view all the answers

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

    <p>It is the point of intersection of the medians</p> Signup and view all the answers

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

    <p>2:1</p> Signup and view all the answers

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

    <p>Centroid</p> Signup and view all the answers

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

    <p>P^2 / 3</p> Signup and view all the answers

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

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

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

    <p>1/2 * (x1<em>y2 - x2</em>y1 + x2<em>y3 - x3</em>y2 + x3<em>y4 - x4</em>y3 + x4<em>y1 - x1</em>y4)</p> Signup and view all the answers

    How can the area of a polygon be obtained?

    <p>By dividing the polygon into triangles and adding their areas</p> 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)?

    <p>1/2 * |x1*(y2 - y3) + x2*(y3 - y1) + x3*(y1 - y2)|</p> 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?

    <p>-4</p> Signup and view all the answers

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

    <p>-4</p> Signup and view all the answers

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

    <p>55</p> Signup and view all the answers

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

    <p>(10, -2)</p> Signup and view all the answers

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

    <p>Incenter</p> Signup and view all the answers

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

    <p>Incenter</p> Signup and view all the answers

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

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

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

    <p>Ratio of remaining sides</p> Signup and view all the answers

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

    <p>(b + c) : a, (c + a) : b, (a + b) : c</p> Signup and view all the answers

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

    <p>Vertices of the triangle</p> Signup and view all the answers

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

    <p>Incircle</p> Signup and view all the answers

    What is the area of the triangle ACD?

    <p>17.5 sq.units</p> Signup and view all the answers

    What is the area of the quadrilateral ABCD?

    <p>28 sq.units</p> Signup and view all the answers

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

    <p>All vertices cannot be rational</p> Signup and view all the answers

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

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

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

    <p>35</p> Signup and view all the answers

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

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

    What is the purpose of the given example?

    <p>To prove that the coordinates of the vertices of an equilateral triangle cannot all be rational</p> Signup and view all the answers

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

    <p>ar (ΔABC) + ar (ΔACD)</p> Signup and view all the answers

    What are the coordinates of the circumcenter of a triangle formed by three points?

    <p>($x_1 sin^2A + x_2 sin^2B + x_3 sin^2C$, $y_1 sin^2A + y_2 sin^2B + y_3 sin^2C$) / (sin^2A + sin^2B + sin^2C)</p> Signup and view all the answers

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

    <p>AP^2 = BP^2 = CP^2</p> Signup and view all the answers

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

    <p>4x - 2y + 5 = 0</p> 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)?

    <p>3/11</p> Signup and view all the answers

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

    <p>6x - 4y + 13 = 0</p> 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)?

    <p>11/3</p> Signup and view all the answers

    What is the circumcenter of the triangle formed by the points A, B, and C?

    <p>The point where the three perpendicular bisectors of the sides of the triangle intersect</p> Signup and view all the answers

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

    <p>AP = BP = CP</p> Signup and view all the answers

    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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    Description

    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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