Question

Прямая, параллельная стороне АС треугольника АВС, пересекает стороны АВ и ВС в точках М и N соответственно, АС = 24, МN = 20. Площадь треугольника МВN равна 75. Найдите площадь треугольника АВС помогите решить срочно пж пж!!:((​

Answer 1

Ответ:

108

Объяснение:

угол МN =углу АВС как соответственные при МN//АС и секущей АВ

Угол АВС общий, значит треугольник ВМN подобен тр-ку АВС .

Коэффициент подобия равен отношению АС к МN , т.е. 20:24=5:6.

Площади подобных треугольников относятся как квадрат коэффициента подобия, т.е. как 25:36.

Площадь треугольника АВС : площадь тр-ка ВМN=36:25

Площадь треугольника АВС = 75*36:25=108

Answer 2

Problem Analysis

We are given a triangle ABC, and a line parallel to side AC that intersects sides AB and BC at points M and N respectively. We are also given that AC = 24, MN = 20, and the area of triangle MVN is 75. We need to find the area of triangle ABC.

Solution

To find the area of triangle ABC, we can use the concept of similar triangles. Since line MN is parallel to side AC, we can conclude that triangles ABC and MVN are similar.

Let's denote the length of side AB as x. Since triangles ABC and MVN are similar, we can set up the following proportion:

AB/MN = AC/BC

Substituting the given values, we have:

x/20 = 24/BC

Simplifying the equation, we get:

x = (20 * 24) / BC x = 480 / BC

Now, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In triangle ABC, the base is AB and the height is the perpendicular distance from point C to line AB. Let's denote this perpendicular distance as h.

The area of triangle ABC is given by:

Area(ABC) = (1/2) * AB * h

To find h, we can use the fact that triangles ABC and MVN are similar. The ratio of corresponding sides in similar triangles is equal. Therefore, we can set up the following proportion:

AB/MN = AC/BC

Substituting the values, we have:

x/20 = 24/BC

Simplifying the equation, we get:

x = (20 * 24) / BC x = 480 / BC

Since triangles ABC and MVN are similar, the ratio of their corresponding heights is also equal. Therefore, we can set up the following proportion:

h/MN = AC/BC

Substituting the values, we have:

h/20 = 24/BC

Simplifying the equation, we get:

h = (20 * 24) / BC h = 480 / BC

Now, we can substitute the values of x and h into the formula for the area of triangle ABC:

Area(ABC) = (1/2) * (480 / BC) * (480 / BC)

Given that the area of triangle MVN is 75, we can set up the following equation:

(1/2) * (20) * (h) = 75

Substituting the value of h, we have:

(1/2) * (20) * (480 / BC) = 75

Simplifying the equation, we get:

(480 / BC) = 7.5

Cross-multiplying, we have:

480 = 7.5 * BC

Dividing both sides by 7.5, we get:

BC = 480 / 7.5 BC = 64

Now, we can substitute the value of BC into the equation for x:

x = 480 / BC x = 480 / 64 x = 7.5

Therefore, the length of side AB is 7.5.

Finally, we can substitute the values of x and BC into the formula for the area of triangle ABC:

Area(ABC) = (1/2) * AB * h Area(ABC) = (1/2) * 7.5 * (480 / 64) Area(ABC) = 3.75 * 7.5 Area(ABC) = 28.125

Therefore, the area of triangle ABC is 28.125.

Answer

The area of triangle ABC is 28.125.