Question

Стороны прямоугольника равны 5 см и 9 см. а) Найдите ширину прямоугольника, равновеликого данному, если его длина равна 15 см. b) В каждом из этих прямоугольников провели диагональ. Будут ли эти прямоугольники равносоставленными? Ответ обоснуйте.

Answer 1

Если прямоугольники равновелики, у них равные площади. Площадь одного из них 45, тогда, зная площадь и длину второго, найдем его ширину. 45/15=3/см/

Если два многоугольника равновелики, то один из них можно разбить  на такие части, из которых возможно составить второй  многоугольник. Они равносоставленные.

а) 3см; б) равносоставленные.

Answer 2

Calculation of the Width of the Rectangle

To find the width of the rectangle, we need to use the formula for the area of a rectangle, which is length multiplied by width. In this case, the length is given as 15 cm and one side of the rectangle is 5 cm. Let's denote the width as w.

Using the formula for the area of a rectangle, we can set up the equation:

15 cm * w = (5 cm) * (9 cm)

Simplifying the equation, we have:

15w = 45

Dividing both sides of the equation by 15, we find:

w = 3 cm

Therefore, the width of the rectangle, which is equal to the given rectangle, is 3 cm.

Comparison of the Rectangles

Now let's consider the diagonals of the two rectangles. The diagonals of a rectangle are equal in length and divide the rectangle into two congruent right triangles.

In the given rectangle with sides measuring 5 cm and 9 cm, we can use the Pythagorean theorem to find the length of the diagonal. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Using the formula, we have:

diagonal^2 = 5^2 + 9^2

Simplifying the equation, we find:

diagonal^2 = 25 + 81

diagonal^2 = 106

Taking the square root of both sides, we find:

diagonal ≈ 10.2956 cm

Now, let's consider the second rectangle, which is equal in area to the given rectangle. Since the area is the same, the length and width of the second rectangle must be different. Let's denote the width of the second rectangle as w2.

Using the formula for the area of a rectangle, we can set up the equation:

15 cm * w2 = (5 cm) * (9 cm)

Simplifying the equation, we have:

15w2 = 45

Dividing both sides of the equation by 15, we find:

w2 = 3 cm

Therefore, the width of the second rectangle is also 3 cm.

Now, let's find the length of the second rectangle using the formula for the area of a rectangle:

15 cm * length2 = (5 cm) * (9 cm)

Simplifying the equation, we have:

15 * length2 = 45

Dividing both sides of the equation by 15, we find:

length2 = 3 cm

Therefore, the length of the second rectangle is also 3 cm.

Since both the width and length of the second rectangle are equal to 3 cm, the second rectangle is also a square.

In conclusion, the two rectangles are not congruent because one is a rectangle with sides measuring 5 cm and 9 cm, while the other is a square with sides measuring 3 cm.