Question

Диагональ делит четырехугольник на два треугольника с периметром 31 см и 41 см .Чему равна длина этой диагонали если периметр четырехугольника равен 42 см ?

Answer 1

пусть диагональ равна х. 2 стороны одного треугольника(без диагонали) 31-х.  2 стороны другого 41-х. Но поскольку эти четыре стороны составляют периметр четырехугольника, то 31-х+41-х=42

72-2х=42

2х=72-42

2х=30

х=15

диагональ равна 15

Answer 2

Problem Analysis

We are given a quadrilateral that is divided by a diagonal into two triangles. The perimeters of these triangles are given as 31 cm and 41 cm. We need to find the length of this diagonal when the perimeter of the quadrilateral is 42 cm.

Solution

Let's assume that the lengths of the sides of the quadrilateral are a, b, c, and d, and the length of the diagonal is x.

We know that the perimeter of a quadrilateral is the sum of the lengths of its sides. So we can write the equation:

a + b + c + d = 42 cm -- (Equation 1)

We are also given that the diagonal divides the quadrilateral into two triangles with perimeters of 31 cm and 41 cm. Let's assume that the lengths of the sides of the triangles formed by the diagonal are p, q, and r for one triangle, and s, t, and u for the other triangle.

We can write the equations for the perimeters of the triangles as:

p + q + x = 31 cm -- (Equation 2) s + t + x = 41 cm -- (Equation 3)

Now, let's solve these equations to find the value of x, the length of the diagonal.

From Equation 2, we can express q in terms of p and x:

q = 31 - p - x

Similarly, from Equation 3, we can express t in terms of s and x:

t = 41 - s - x

Now, let's substitute these expressions into Equation 1 to eliminate q and t:

a + b + c + d = p + (31 - p - x) + s + (41 - s - x)

Simplifying this equation, we get:

a + b + c + d = 72 - x

Since we know that a + b + c + d = 42 cm (from Equation 1), we can substitute this value into the equation above:

42 = 72 - x

Solving for x, we find:

x = 72 - 42 x = 30 cm

Therefore, the length of the diagonal is 30 cm.

Conclusion

The length of the diagonal that divides the quadrilateral into two triangles with perimeters of 31 cm and 41 cm, when the perimeter of the quadrilateral is 42 cm, is 30 cm.