Question

площадь прямоугольного треугольника abc (угол bac= 90 градусов) равна 24 см2 , ac = 8 см .Отрезок mf- биссектриса прямоугольного треугольника mkp ( угол mkp = 90 градусов), mp=25 см,kf:fp=3:5. Подобны ли треугольники abc и kmp ? Ответ обоснуйте

Answer 1

Рассмотрим треугольник ABC. Зная площадь и катет найдем другой катет AB=6 см.
Рассмотрим треугольник KMP. Зная что MK:MP=3:5 и MP=25 находим что MK=15. По теореме Пифагора KP=20.
Если считать соответственными сторонами пары KP и АС, KM и АВ , то получится два отношения 20/8 и 15/6, которые равны >> треугольники abc и kmp подобны по второму признаку подобия треугольников(прямой угол у них общий) 

Answer 2

Calculation of the Area of Triangle ABC

To find the area of triangle ABC, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the base of triangle ABC is AC and the height is BC. Given that AC = 8 cm and the area of triangle ABC is 24 cm², we can substitute these values into the formula to find the height BC:

24 = (1/2) * 8 * BC

Simplifying the equation, we have:

48 = 8 * BC

Dividing both sides of the equation by 8, we find:

BC = 6 cm

Therefore, the height of triangle ABC is 6 cm.

Calculation of the Lengths of Segments MF and FP

To determine if triangles ABC and KMP are similar, we need to compare their corresponding sides. Given that MP = 25 cm and the ratio of KF to FP is 3:5, we can calculate the lengths of segments MF and FP.

Let's assume that KF is 3x and FP is 5x, where x is a constant. Since the ratio of KF to FP is 3:5, we can set up the following equation:

3x + 5x = MP

Simplifying the equation, we have:

8x = 25

Dividing both sides of the equation by 8, we find:

x = 25/8

Therefore, KF is equal to 3 times 25/8, and FP is equal to 5 times 25/8.

Comparison of Triangles ABC and KMP

To determine if triangles ABC and KMP are similar, we need to compare their corresponding sides. The sides of triangle ABC are AB, BC, and AC, while the sides of triangle KMP are KM, MP, and KP.

Comparing the sides of the two triangles, we have:

- AB is not equal to KM, as they are different sides. - BC is not equal to MP, as they are different sides. - AC is not equal to KP, as they are different sides.

Since none of the corresponding sides are equal, triangles ABC and KMP are not similar.

Therefore, the triangles ABC and KMP are not similar.