Question

В выпуклом четырхугольнике АВСD диагонали АС и ВD пересекаются в точке О, причём угол ОВС = углу ОDA; ВО = ОD. Периметр треугольника ВОС равен 26 см, а периметр треугольника АОВ равен 32 см; АD = 10 см.1) Докажите,что четырёхугольник АВСD - параллелограмм2) Найдите периметр четырёхугольника АВСD

Answer 1

1) ВС II АД, т.к. уголОВС=углуОДА, а они накрест лежащие при ВС и АД и секущей ВД.
треугольникВОС=треугольникуАОД по 2 признаку (ВО=ОД по условию, уголОВС=углуОДА по условию, уголВОС=углуАОД т.к. вертикальные). Следовательно, ВС=АД, АО=ОС.
Если в четырехугольнике две стороны равны и параллельны, то такой четырехугольник - параллелограмм.
У нас ВС II АД и ВС=АД, следовательно, АВСД - параллелограмм.
2) ВО+ОС=26-10=16см
ВО+ОС=ВО+АО=16см
АВ=32-16=16см
Равсд=(16+10)*2=52см.

Answer 2

1) Докажите, что четырёхугольник ABCD - параллелограмм:

To prove that quadrilateral ABCD is a parallelogram, we need to show that opposite sides are parallel and equal in length.

Given that the diagonals AC and BD intersect at point O, and angle OVS is equal to angle ODA, we can use this information to prove that the opposite sides are parallel.

Let's start by proving that side AB is parallel to side CD.

We know that angle OVS is equal to angle ODA, and by the property of alternate interior angles, we can conclude that angle OAB is equal to angle OCD. This is because angles OVS and OAB are corresponding angles, and angles ODA and OCD are also corresponding angles.

Therefore, we can say that angle OAB is congruent to angle OCD.

Now, let's consider side AD. We know that OV is parallel to AD because angle OVS is equal to angle ODA.

Since OV is parallel to AD, and AB is parallel to CD, we can conclude that side AB is parallel to side CD.

Similarly, we can prove that side BC is parallel to side AD using the same logic.

Hence, we have proved that opposite sides AB and CD are parallel.

Now, let's prove that opposite sides AD and BC are parallel.

We know that angle OAB is equal to angle OCD. By the property of alternate interior angles, we can conclude that angle ODA is equal to angle OBC. This is because angles OAB and ODA are corresponding angles, and angles OCD and OBC are also corresponding angles.

Therefore, we can say that angle ODA is congruent to angle OBC.

Since angle ODA is congruent to angle OBC, and angle OVS is equal to angle ODA, we can conclude that angle OVS is congruent to angle OBC.

Since OV is parallel to AD, and angle OVS is congruent to angle OBC, we can say that side AD is parallel to side BC.

Hence, we have proved that opposite sides AD and BC are parallel.

Since we have proved that both pairs of opposite sides are parallel, we can conclude that quadrilateral ABCD is a parallelogram.

2) Найдите периметр четырёхугольника ABCD:

To find the perimeter of quadrilateral ABCD, we need to find the lengths of all four sides.

We are given that the perimeter of triangle BOC is 26 cm, and the perimeter of triangle AOB is 32 cm.

Let's denote the lengths of the sides as follows: - Side AB = x - Side BC = y - Side CD = z - Side DA = w

From the given information, we can write two equations based on the perimeters of the triangles:

Equation 1: x + y + z = 26 cm Equation 2: w + x + y = 32 cm

Since quadrilateral ABCD is a parallelogram, opposite sides are equal in length. Therefore, we can write the following equations:

Equation 3: x = z Equation 4: y = w

Substituting equations 3 and 4 into equations 1 and 2, we get:

Equation 5: z + y + z = 26 cm Equation 6: w + x + y = 32 cm

Simplifying equations 5 and 6, we obtain:

Equation 7: 2z + y = 26 cm Equation 8: w + x + y = 32 cm

Since we have four variables and only two equations, we need additional information to solve for the lengths of the sides.

Please provide any additional information or values given in the problem statement, and I will be able to assist you further in finding the perimeter of quadrilateral ABCD.