Question

Точки М и N лежат на сторонах АС и ВС треугольника АВС соответственно AC = 16, BC= 12, CN= 9, CM = 12. Докожите что MN||AB. ​

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

Given Information

The points M and N lie on the sides AC and BC of triangle ABC respectively, with AC = 16, BC = 12, CN = 9, and CM = 12.

Proof that MN is Parallel to AB

To prove that MN is parallel to AB, we can use the concept of similar triangles and the property that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.

1. Using Similar Triangles: - Triangles AMC and BNC are similar because of the Angle-Angle (AA) criterion. - This means that the corresponding sides of these triangles are proportional.

2. Using Proportional Sides: - From the given information, we can calculate the lengths of AM and BN using the concept of proportional sides in similar triangles.

3. Concluding Parallelism: - By showing that the ratio of lengths AM/AC is equal to the ratio of lengths BN/BC, we can conclude that MN is parallel to AB.

Let's proceed with the calculations.

Calculation

1. Using Similar Triangles: - Triangles AMC and BNC are similar due to the AA criterion, as angle AMC = angle BNC (both are right angles), and angle CAM = angle CBN (both are common angles).

2. Using Proportional Sides: - Using the property of similar triangles, we can set up the following proportion: - AM/AC = CN/BC - BN/BC = CM/AC

- Substituting the given values: - AM/16 = 9/12 - BN/12 = 12/16

- Solving for AM and BN: - AM = (9/12) * 16 = 12 - BN = (12/16) * 12 = 9

3. Concluding Parallelism: - Since AM = CM and BN = CN, and AM/AC = CN/BC and BN/BC = CM/AC, we can conclude that MN is parallel to AB.

Therefore, MN is parallel to AB.

This proves that the line MN is parallel to the side AB of triangle ABC.