Trapezoid - information

Trapezoid – a quadrilateral having at least one pair of parallel sides; The (selected) pair of parallel sides are called the bases, the remaining sides are called the legs, the distance between the bases is called the height of the trapezoid. Some colloquial definitions define a trapezoid as a quadrilateral with only one pair of parallel sides and, accordingly, the parallelogram is not a trapezoid. The sum of the angles on the same legs is 180°. If P is the intersection of the diagonals, then it's triangles ΔADP and ΔBCP have equal fields, ΔABP and ΔCDP are similar. Triangles ΔABC and ΔABD have a common base and equal height, and therefore equal fields. Triangles ΔBCP and ΔADP they arise from them by "subtracting" the triangle ΔABP.

Trapezoid has the following properties:

1. Trapezoid is a convex figure.
2. The sum of the measures of all interior angles is 2Π (360 °), and the sum of the measures of two adjacent internal angles lying on the same leg is Π, $$ \alpha + \delta = 180° $$$$ \beta + \gamma = 180° $$
3. Formula for the first diagonal of the trapezoid from the sides (a) and (b) and angle β
$$ e=\sqrt{a^2+b^2-2ab\cdot \cos(\beta)} $$
4. Formula for the first diagonal of the trapezoid from sides (c) i (d) and angle δ
$$ e=\sqrt{c^2+d^2-2cd\cdot \cos(\delta)} $$
5. Formula for the first diagonal of the trapezoid from sides (a)(b)(c)(d)
$$ e= \sqrt{\frac{ac^2 - a^2c - ad^2 + b^2c}{c-a}} $$
6. Formula for the second diagonal of the trapezoid from sides (a) i (d) and angle α
$$ f=\sqrt{a^2+d^2-2ad\cdot \cos(\alpha)} $$
7. Formula for the second diagonal of the trapezoid from sides (b) i (c) and angle γ
$$ f=\sqrt{b^2+c^2-2bc\cdot \cos(\gamma)} $$
8. Formula for the second diagonal of the trapezoid from sides (a)(b)(c)(d)
$$ f= \sqrt{\frac{ac^2 - a^2c - ab^2 + cd^2}{c-a}} $$
9. Formula for height of the trapezoid from the sides (a)(c) and area
$$ h=\frac{2S}{a+c} $$
10. Formula for height of the trapezoid from the sides (b)(d) and angle
$$ h=b \cdot \sin \beta = b \cdot \sin \gamma $$ $$ = d \cdot \sin \alpha = d \cdot \sin \delta $$
11. Formula for height of the trapezoid from the sides
$$ h= \frac{\sqrt{i \cdot j \cdot k \cdot l}}{2|c-a|} $$ where: $$ i=-a+b+c+d,$$$$j=a+b-c+d,$$$$k=a+b-c-d,$$$$l=a-b-c+d.$$
12. Formula for area of the trapezoid from sides (a) and (b) and the height (h)
$$ S={\frac {a+c}{2}}\cdot h $$
13. Formula for trapezoidal area from sides and angle (α) or angle(β)
$$ S=\frac{1}{2}\cdot\left(a+c\right)\cdot d\cdot \sin\beta $$$$ S=\frac{1}{2}\cdot\left(a+c\right)\cdot b\cdot \sin\beta $$
14. Formula for area of the trapezoid from the sides (a)(b)(c)(d)
$$ S=\frac{a+c}{4|c-a|}\sqrt{i \cdot j \cdot k \cdot l}. $$ where: $$ i=-a+b+c+d,$$$$j=a+b-c+d,$$$$k=a+b-c-d,$$$$l=a-b-c+d.$$
15. Wzór na Perimeter of the trapezoid from the sides
$$ L = a + b + c + d $$
16. Formula for perimeter of a trapezoid from bases, heights and angles (α) and (β)
$$ L = a+c+h*(\frac{1}{\sin(\alpha)}+\frac{1}{\sin(\beta)}) $$
17. Formula for lenght side (a) from sides (b)(c)(d) and height (h)
$$ a=c+\sqrt{d^2-h^2}+\sqrt{b^2-h^2} $$
18. Formula for lenght leg (b) from sides (a)(c)(d) and height (h)
$$ b=\sqrt{h^2-(-a+c+\sqrt{d^2-h^2})^2} $$
19. Formula for lenght side (c) from sides (a)(b)(d) and height (h)
$$ c=a-\sqrt{d^2-h^2}-\sqrt{b^2-h^2} $$
20. Formula for lenght leg (d) from sides (a)(b)(c) and height (h)
$$ d=\sqrt{h^2+(a-c-\sqrt{b^2-h^2})^2} $$
21. Formula for the midline (median , midsegment) of the trapezoid $$ m=\frac {a+c}{2} $$