Question

Прошу! В треугольник с основанием AC=11 см и высотой BD=8 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

Пусть сторона квадрата равна x. Треугольники ABC и LBM подобны, поэтому отношения сходственных элементов этих треугольников равны. Высота второго Δ равна 8-x, поэтому AC/LM=8/(8-x); 11/x=8/(8-x); 
88-11x=8x; 19x=88; x=88/19

Ответ: 88/19

Answer 2

Problem Analysis

We are given a triangle ABC with base AC measuring 11 cm and height BD measuring 8 cm. Inside this triangle, a square KLMN is inscribed such that side KN lies on the base AC, and vertices L and M lie on sides AB and BC respectively. We need to determine the length of the side of the square.

Solution

To find the length of the side of the square, we can use the properties of similar triangles. Let's analyze the given information and apply the relevant concepts.

1. Triangle ABC is a right triangle with base AC and height BD. We can use the Pythagorean theorem to find the length of side AB: - AC = 11 cm - BD = 8 cm

Applying the Pythagorean theorem: AB^2 + BD^2 = AC^2 AB^2 + 8^2 = 11^2 AB^2 + 64 = 121 AB^2 = 121 - 64 AB^2 = 57 AB = √57 cm

2. Triangle ABC and triangle KBN are similar triangles. This is because angle KBN is common to both triangles, and angle ABC is a right angle, making angle KBC congruent to angle KBN. Therefore, we can set up the following proportion:

AB/AC = KN/BC

Substituting the known values: √57/11 = KN/BC

Solving for KN: KN = (√57/11) * BC

3. Since the square KLMN is inscribed in triangle ABC, we know that the side KN of the square is parallel to the base AC. Therefore, the length of side KN is equal to the length of the base AC.

KN = AC

Substituting the value of KN from step 2: AC = (√57/11) * BC

4. Now, we can solve for the length of the side of the square by substituting the value of AC from step 3 into the equation:

(√57/11) * BC = AC

Simplifying the equation: (√57/11) * BC = (√57/11) * 11 BC = 11

Therefore, the length of the side of the square is 11 cm.

Answer

The length of the side of the square is 11 cm.