The expression for the area of the triangle in terms of the coordinates of its vertices can thus be given as, Similarly, the bases and heights of the other two trapeziums can be easily calculated. BD and AE can easily be seen to be the y coordinates of B and A, while DE is the difference between the x coordinates of A and B. Its bases are BD and AE, and its height is DE. Trapezium Area = (1/2) × Sum of bases × HeightĬonsider any one trapezium, say BAED. Now, the area of a trapezium in terms of the lengths of the parallel sides (the bases of the trapezium) and the distance between the parallel sides (the height of the trapezium): We can express the area of a triangle in terms of the areas of these three trapeziums.Īrea(ΔABC) = Area(Trap.BAED) + Area(Trap.ACFE) - Area(Trap.BCFD) Notice that three trapeziums are formed: BAED, ACFE, and BCFD. In this figure, we have drawn perpendiculars AE, CF, and BD from the vertices of the triangle to the horizontal axis. Consider ΔABC as given in the figure below with vertices A(x 1, y 1), B(x 2, y 2), and C(x 3, y 3). In coordinate geometry, if we need to find the area of a triangle, we use the coordinates of the three vertices. Proof of Area of Triangle Formula in Coordinate Geometry Let us learn more about it in the following section. If (x 1, y 1), (x 2, y 2), and (x 3, y 3) are the three vertices of a triangle on the coordinate plane, then its area is calculated by the formula (1/2) |x 1(y 2 − y 3) + x 2(y 3 − y 1) + x 3(y 1 − y 2)|. Area of a Triangle Formula in Coordinate Geometry Now, with the help of coordinate geometry, we can find the area of this triangle. The area covered by the triangle ABC in the x-y plane is the region marked in blue. If you plot these three points in the plane, you will find that they are non-collinear, which means that they can be the vertices of a triangle, as shown below: Let us understand the concept of the area of a triangle in coordinate geometry better using the example given below,Ĭonsider these three points: A(−2,1), B(3,2), C(1,5). The area of a triangle in coordinate geometry is defined as the area or space covered by it in the 2-D coordinate plane. The area of a triangle in coordinate geometry can be calculated if the three vertices of the triangle are given in the coordinate plane. Integrating curvatures over beam length, the deflection, at some point along x-axis, should also be reversely proportional to I.What is the Area of a Triangle in Coordinate Geometry?Ĭoordinate geometry is defined as the study of geometry using coordinate points. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the developed curvature is reversely proportional to the moment of inertia I. Thus their combined moment of inertia is: These triangles, have common base equal to h, and heights b1 and b2 respectively. The moment of inertia of a triangle with respect to an axis perpendicular to its base, can be found, considering that axis y'-y' in the figure below, divides the original triangle into two right ones, A and B. This can be proved by application of the Parallel Axes Theorem (see below) considering that triangle centroid is located at a distance equal to h/3 from base. The moment of inertia of a triangle with respect to an axis passing through its base, is given by the following expression: Where b is the base width, and specifically the triangle side parallel to the axis, and h is the triangle height (perpendicular to the axis and the base). The moment of inertia of a triangle with respect to an axis passing through its centroid, parallel to its base, is given by the following expression:
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