what is the property of trapezium
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The chief properties of the Trapezium are as follow:
1. Any three sides of a Trapezium taken together, are greater than the third side.
2. The two diagonals of any Trapezium divide it into four proportional triangles, a, b, c, d. That is, the triangle a : b :: c : d.
3. The sum of all the four inward angles, A, B, C, D, taken together, is equal to 4 right angles, or 360°.
4. In a Trapezium ABCD, if all the sides be bisected, in the points E, F, G, H, the figure EFGH formed by joining the points of bisection will be a parallelogram, having its opposite sides parallel to the corresponding diagonals of the Trapezium, and the area of the said inscribed parallelogram is just equal to half the area of the Trapezium.
5. The sum of the squares of the diagonals of the Trapezium, is equal to twice the sum of the squares of the diagonals of the parallelogram, or of the two lines drawn to bisect the opposite sides of the Trapezium. That is, .
6. In any Trapezium, the sum of the squares of all the four sides, is equal to the sum of the squares of the two diagonals together with 4 times the square of the line KI joining their middle points. That is, (first fig. below) .
7. In any Trapezium, the sum of the two diago- nals, is less than the sum of any four lines that can be drawn, to the four angles, from any point within the figure, beside the intersection of the diagonals.
8. The area of any Trapezium, is equal to half the rectangle or product under either diagonal and the sum of the two perpendiculars drawn upon it from the two opposite angles.
9. The area of any Trapezium may also be found thus: Multiply the two diagonals together, then that product, multiplied by the sine of their angle of intersection, to the radius 1, will be the area. That is, .
10. The same area will be otherwise found thus: Square each side of a Trapezium, add the squares of each pair of opposite sides together, subtract the less sum from the greater, multiply the remainder by the tangent of the angle of intersection of the diagonals (to radius 1), and 1/4 of the product will be the area. That is, .
11. The area of a Trapezoid, or one that has two sides parallel, is equal to the rectangle or product under the sum of the two parallel sides and the perpendicular distance between them.
12. If a Trapezium be inscribed in a circle; the sum of any two opposite angles is equal to two right angles; and if the sum of two opposite angles be equal to two right angles, the sum of the other two will also be equal to two right angles, and a circle may be described about it; and farther, if one side, as DC, be produced out, the external angle will be equal to the interior opposite angle. That is, (last fig. above) .
13. If a Trapezium be inscribed in a circle; the rectangle of the two diagonals, is equal to the sum of the two rectangles contained under the opposite sides. That is, .
14. If a Trapezium be inscribed in a circle; its area may be found thus: Multiply any two adjacent sides together, and the other two sides together; then add these two products together, and multiply the sum by the sine of the angle included by either of the pairs of sides that are multiplied together, and half this last product will be the area. That is, the area is equal either .
15. Or, when the Trapezium can be inscribed in a circle, the area may be otherwise found thus: Add all the four sides together, and take half the sum; then from this half subtract each side severally; multiply the four remainders continually together, and the square root of the last product will be the area.
16. Lastly, the area of the Trapezium inscribed in a circle may be otherwise found thus: Put,,,, then (√mnp)/4r = the area of the Trapezium.
1. Any three sides of a Trapezium taken together, are greater than the third side.
2. The two diagonals of any Trapezium divide it into four proportional triangles, a, b, c, d. That is, the triangle a : b :: c : d.
3. The sum of all the four inward angles, A, B, C, D, taken together, is equal to 4 right angles, or 360°.
4. In a Trapezium ABCD, if all the sides be bisected, in the points E, F, G, H, the figure EFGH formed by joining the points of bisection will be a parallelogram, having its opposite sides parallel to the corresponding diagonals of the Trapezium, and the area of the said inscribed parallelogram is just equal to half the area of the Trapezium.
5. The sum of the squares of the diagonals of the Trapezium, is equal to twice the sum of the squares of the diagonals of the parallelogram, or of the two lines drawn to bisect the opposite sides of the Trapezium. That is, .
6. In any Trapezium, the sum of the squares of all the four sides, is equal to the sum of the squares of the two diagonals together with 4 times the square of the line KI joining their middle points. That is, (first fig. below) .
7. In any Trapezium, the sum of the two diago- nals, is less than the sum of any four lines that can be drawn, to the four angles, from any point within the figure, beside the intersection of the diagonals.
8. The area of any Trapezium, is equal to half the rectangle or product under either diagonal and the sum of the two perpendiculars drawn upon it from the two opposite angles.
9. The area of any Trapezium may also be found thus: Multiply the two diagonals together, then that product, multiplied by the sine of their angle of intersection, to the radius 1, will be the area. That is, .
10. The same area will be otherwise found thus: Square each side of a Trapezium, add the squares of each pair of opposite sides together, subtract the less sum from the greater, multiply the remainder by the tangent of the angle of intersection of the diagonals (to radius 1), and 1/4 of the product will be the area. That is, .
11. The area of a Trapezoid, or one that has two sides parallel, is equal to the rectangle or product under the sum of the two parallel sides and the perpendicular distance between them.
12. If a Trapezium be inscribed in a circle; the sum of any two opposite angles is equal to two right angles; and if the sum of two opposite angles be equal to two right angles, the sum of the other two will also be equal to two right angles, and a circle may be described about it; and farther, if one side, as DC, be produced out, the external angle will be equal to the interior opposite angle. That is, (last fig. above) .
13. If a Trapezium be inscribed in a circle; the rectangle of the two diagonals, is equal to the sum of the two rectangles contained under the opposite sides. That is, .
14. If a Trapezium be inscribed in a circle; its area may be found thus: Multiply any two adjacent sides together, and the other two sides together; then add these two products together, and multiply the sum by the sine of the angle included by either of the pairs of sides that are multiplied together, and half this last product will be the area. That is, the area is equal either .
15. Or, when the Trapezium can be inscribed in a circle, the area may be otherwise found thus: Add all the four sides together, and take half the sum; then from this half subtract each side severally; multiply the four remainders continually together, and the square root of the last product will be the area.
16. Lastly, the area of the Trapezium inscribed in a circle may be otherwise found thus: Put,,,, then (√mnp)/4r = the area of the Trapezium.
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