Given that ABC is a right triangle and line CD is perpendicular to line AB, which proportions are true? Select the two correct answers.
1. AC/BC=AD/DB
2. BC/AD=AB/AC
3. BC/DC=AB/BC
4. AC/AD=AB/AC
5. BC/DB=AB/BC
To find the correct proportions, we can use the properties of right triangles and perpendicular lines.
Let's label the points first: A, B, C, and D.
Since line CD is perpendicular to line AB and point D forms a right angle, we have a right triangle with AC as the hypotenuse.
Now, let's examine the statements one by one:
1. AC/BC = AD/DB: This proportion is true. In a right triangle, the sides AC and BC are the two legs of the triangle, while AD and DB are the segments of the hypotenuse. By dividing both sides, we get the ratio of the lengths of the legs.
2. BC/AD = AB/AC: This proportion is also true. By rearranging the terms, we see that it is equivalent to the statement of the Pythagorean Theorem. In a right triangle, the sum of the squares of the lengths of the legs (BC and AD) is equal to the square of the hypotenuse (AB).
3. BC/DC = AB/BC: This proportion is not correct. The line segment DC does not provide sufficient information to relate it to the other sides of the triangle. Therefore, we cannot establish a proportional relationship between them.
4. AC/AD = AB/AC: This proportion is not correct. It violates the Pythagorean Theorem because AC, AD, and AB form a right triangle. According to the theorem, the sum of the squares of the legs is equal to the square of the hypotenuse, not the other way around.
5. BC/DB = AB/AC: This proportion is not correct. It violates the Pythagorean Theorem, as BC, DB, and AB form a right triangle.
Therefore, the correct proportions are:
1. AC/BC = AD/DB
2. BC/AD = AB/AC
Answers: (1) and (2)
CD divides ABC into two similar triangles, that are also similar to ABC itself
ABC ~ CBD ~ ACD