Question

В треугольник с основанием AC=18 см и высотой BD=5 см вписан квадрат KLMN так, что сторона KN лежит на основании AC, а вершины L и M — соответственно на сторонах AB и BC. Определи длину стороны квадрата. (Ответ запиши в виде не сокращённой дроби.) Длина стороны квадрата равна...?

Answer 1

Відповідь:

ответ на фото.

Answer 2

Task: Find the length of the side of the square inscribed in a triangle.

To find the length of the side of the square inscribed in the triangle, we can use the properties of similar triangles.

Let's analyze the given information: - The triangle ABC has a base AC with a length of 18 cm and a height BD with a length of 5 cm. - The square KLMN is inscribed in the triangle, with side KN lying on the base AC, and vertices L and M on the sides AB and BC, respectively.

To find the length of the side of the square, we need to determine the relationship between the side of the square and the base of the triangle.

Solution:

1. Let's denote the length of the side of the square as x cm. 2. Since the square is inscribed in the triangle, the side KN of the square is parallel to the base AC of the triangle. 3. According to the properties of similar triangles, we can set up the following proportion:

KN / AC = BD / AB

Substituting the given values:

x / 18 = 5 / AB

4. Solving for AB, we can rewrite the equation as:

AB = (18 * 5) / x

5. Now, we need to find the relationship between AB and BC. Since the triangle ABC is a right triangle, we can use the Pythagorean theorem:

AB^2 + BC^2 = AC^2

Substituting the values:

(AB)^2 + (BC)^2 = 18^2

Simplifying:

(AB)^2 + BC^2 = 324

6. Substituting the value of AB from step 4:

[(18 * 5) / x]^2 + BC^2 = 324

7. Simplifying further:

(18 * 5)^2 / x^2 + BC^2 = 324

8. Rearranging the equation:

BC^2 = 324 - [(18 * 5)^2 / x^2]

9. Now, we know that the side KN of the square is parallel to the base AC, so the length of BC is equal to the length of KN, which is x cm.

BC = x

10. Substituting BC = x into the equation from step 8:

x^2 = 324 - [(18 * 5)^2 / x^2]

11. Simplifying further:

x^4 = 324x^2 - (18 * 5)^2

12. Rearranging the equation:

x^4 - 324x^2 + (18 * 5)^2 = 0

13. This is a quadratic equation in terms of x^2. We can solve it using the quadratic formula:

x^2 = (-b ± √(b^2 - 4ac)) / (2a)

where a = 1, b = -324, and c = (18 * 5)^2.

14. Substituting the values into the quadratic formula:

x^2 = (-(-324) ± √((-324)^2 - 4 * 1 * (18 * 5)^2)) / (2 * 1)

15. Simplifying further:

x^2 = (324 ± √(104976 - 4 * 1 * 1620)) / 2

16. Continuing to simplify:

x^2 = (324 ± √(104976 - 6480)) / 2

x^2 = (324 ± √(98796)) / 2

17. Taking the positive square root:

x = √(98796) / 2

18. Evaluating the square root:

x ≈ 99.397 / 2

x ≈ 49.6985

19. Therefore, the length of the side of the square is approximately 49.6985 cm.

Please note that the answer is rounded to four decimal places.

Let me know if you need any further assistance!