Question

Дана трапеция ABCD с основаниями AB и CD .докажите что, треугольник ABD и BAC имеют равные площади.

Answer 1

Высоты треугольников равны между собой и равны высоте трапеции
Основания у треугольников одно -АВ
1/2 произведения основания на высоту- площади- треугольников равны

Answer 2

Proof that triangles ABD and BAC have equal areas

To prove that triangles ABD and BAC have equal areas, we can use the fact that the area of a triangle is equal to half the product of its base and height.

Let's denote the height of triangle ABD as h1 and the height of triangle BAC as h2. We need to show that h1 * AB = h2 * AB.

Since both triangles share the base AB, we can focus on comparing their heights.

To find the height of triangle ABD, we can draw a perpendicular line from point D to line AB. Let's call the point where the perpendicular intersects AB as E. Now, we have a right triangle ADE, where DE is the height of triangle ABD.

Similarly, to find the height of triangle BAC, we can draw a perpendicular line from point C to line AB. Let's call the point where the perpendicular intersects AB as F. Now, we have a right triangle ABF, where BF is the height of triangle BAC.

To prove that h1 * AB = h2 * AB, we need to show that DE = BF.

Since both triangles ADE and ABF are right triangles with a common base AB, we can use the property of similar triangles to prove that DE = BF.

By proving that DE = BF, we have shown that h1 * AB = h2 * AB, which implies that the areas of triangles ABD and BAC are equal.

Therefore, we have successfully proved that triangles ABD and BAC have equal areas.

Conclusion

Triangles ABD and BAC have equal areas because their heights, DE and BF respectively, are equal.