Question

К плоскости треуголтника со сторонами 26 см,28 см,30 см из вершины среднего угла проведён перпендикуляр длиной 32 см.Найдите расстояние от концов перпендикуляра до противоположной стороны

Answer 1

Сначала для облегчения вычислений уменьшим все данные в два раза, в результате и требуемое расстояние уменьшится в два раза. Закончив вычисления, нужно будет не забыть полученное расстояние в два раза увеличить. Итак, берем стороны треугольника 13, 14, 15 и перпендикуляр 16. Против среднего угла лежит средняя сторона, ее длина равна 14. Найдем сначала высоту треугольника, опущенную на сторону длиной 14. Для этого сначала найдем площадь треугольника по формуле Герона
S=$\sqrt{p(p-a)(p-b)(p-c)}$, где a; b; c - стороны треугольника, а p - полупериметр: S=$\sqrt{21·6·8·7}$=7·3·4=84. С другой стороны, площадь треугольника равна половине произведения основания 14 на высоту h, откуда h=84·2/14=12. Далее нужно воспользоваться теоремой о трех перпендикулярах, из которой следует, что требуемый отрезок равен 
$\sqrt{[tex] 16^{2}+[tex] 12^{2}$ [/tex]} [/tex]=$\sqrt{400}$=20. Не забываем в конце домножить результат на 2.
Ответ: 40

Answer 2

Task: Finding the distance from the ends of a perpendicular drawn from the vertex of the median angle to the opposite side of a triangle.

To find the distance from the ends of a perpendicular drawn from the vertex of the median angle to the opposite side of a triangle, we can use the concept of similar triangles.

Let's break down the problem step by step:

1. Given information: - The triangle has sides of lengths 26 cm, 28 cm, and 30 cm. - A perpendicular is drawn from the vertex of the median angle, and its length is 32 cm.

2. Finding the median angle: - In a triangle, the median angle is the angle formed by the median and the side opposite to it. - The median divides the triangle into two smaller triangles with equal areas. - The median angle is the angle formed by the two sides of the triangle that are divided by the median.

3. Finding the length of the median: - The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. - In this case, the median is the line segment connecting the vertex of the median angle to the midpoint of the side opposite to it. - To find the length of the median, we can use the formula: median = (1/2) * sqrt(2 * (a^2 + b^2) - c^2), where a, b, and c are the lengths of the sides of the triangle. - Substituting the given values, we have: median = (1/2) * sqrt(2 * (26^2 + 28^2) - 30^2). - Calculating the value of the median will give us the length of the line segment connecting the vertex of the median angle to the midpoint of the opposite side.

4. Finding the ratio of the lengths: - Since the two triangles formed by the median are similar, the ratio of the lengths of corresponding sides is the same. - Let's denote the distance from one end of the perpendicular to the opposite side as x, and the distance from the other end of the perpendicular to the opposite side as y. - We can set up the following proportion: x / y = median / perpendicular. - Substituting the values we have, we get: x / y = median / 32.

5. Solving for x and y: - We can rearrange the proportion to solve for x and y: x = (median / 32) * y. - Since x and y represent the distances from the ends of the perpendicular to the opposite side, their sum will be equal to the length of the opposite side. - Therefore, we can write the equation: x + y = opposite side. - Substituting the value of x from the rearranged proportion, we get: (median / 32) * y + y = opposite side. - Simplifying the equation, we have: ((median + 32) / 32) * y = opposite side. - Solving for y, we get: y = (32 * opposite side) / (median + 32). - Substituting the value of y back into the equation, we can find the value of x: x = (median / 32) * ((32 * opposite side) / (median + 32)).

6. Calculating the final values: - Substitute the values of the median and the opposite side into the equations to find the values of x and y. - Calculate the sum of x and y to find the distance from the ends of the perpendicular to the opposite side.

Please note that the exact numerical values for the median, opposite side, x, and y will depend on the specific lengths of the sides of the triangle given in the problem.

Let me know if you have any further questions!