Question

В выпуклом четырехугольнике АВСД, АВ=9см, ВС=8см, СД=16см, АД=6см, ВД=12см. Докажите, что АВСД-трапеция.»с объяснением

Answer 1

Докажем, что AB || CD, а для этого достаточно доказать, что углы BDC и DBA равны. Для этого применим теорему косинусов к треугольникам BDC и ABD. В одном стороны равны 8, 12, 16 против угла BDC лежит сторона длиной 8, в другом - 9, 6, 12, против угла ABD лежит сторона длиной 6. Косинусы обоих углов будут равны 7/8  (просто подставляем числа в формулу и считаем), а из этого следует равенство углов и параллельность прямых.

Answer 2

Given Information

We are given a convex quadrilateral ABCD, where AB = 9 cm, BC = 8 cm, CD = 16 cm, AD = 6 cm, and BD = 12 cm. We need to prove that ABCD is a trapezoid.

Proof

To prove that ABCD is a trapezoid, we need to show that one pair of opposite sides is parallel.

Let's consider the given information and use it to prove that AB || CD.

According to the given information, AB = 9 cm, BC = 8 cm, CD = 16 cm, AD = 6 cm, and BD = 12 cm.

Using the fact that the opposite sides of a parallelogram are equal, we can say that AB = CD.

Now, let's consider the triangles ABD and BCD.

In triangle ABD, we have AB = 9 cm, AD = 6 cm, and BD = 12 cm.

In triangle BCD, we have BC = 8 cm, CD = 16 cm, and BD = 12 cm.

Using the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In triangle ABD, we have AB + AD > BD, which becomes 9 + 6 > 12, which is true.

In triangle BCD, we have BC + CD > BD, which becomes 8 + 16 > 12, which is also true.

Therefore, both triangles ABD and BCD are valid triangles.

Now, let's consider the angles of triangles ABD and BCD.

In triangle ABD, we have angle ABD.

In triangle BCD, we have angle BCD.

Using the fact that the sum of the angles in a triangle is 180 degrees, we can say that angle ABD + angle BCD = 180 degrees.

Now, let's consider the angles of quadrilateral ABCD.

In quadrilateral ABCD, we have angle ABD, angle BCD, angle BDC, and angle CAD.

Using the fact that the sum of the angles in a quadrilateral is 360 degrees, we can say that angle ABD + angle BCD + angle BDC + angle CAD = 360 degrees.

Since angle ABD + angle BCD = 180 degrees, we can rewrite the equation as 180 degrees + angle BDC + angle CAD = 360 degrees.

Simplifying the equation, we get angle BDC + angle CAD = 180 degrees.

Now, let's consider the angles of triangle BDC.

In triangle BDC, we have angle BDC.

Using the fact that the sum of the angles in a triangle is 180 degrees, we can say that angle BDC + angle BCD + angle CAD = 180 degrees.

Since angle BDC + angle CAD = 180 degrees, we can rewrite the equation as angle BCD = angle BDC.

Therefore, we have angle BCD = angle BDC.

Using the fact that corresponding angles are congruent when two lines are cut by a transversal, we can say that AB || CD.

Hence, we have proved that ABCD is a trapezoid.

Conclusion

We have shown that in the given convex quadrilateral ABCD, AB || CD. Therefore, ABCD is a trapezoid.

Note: The proof provided above is based on the given information and the properties of triangles and quadrilaterals.