Aaron is proving that the slope between any two points on a straight line is the same. He has already proven that triangles
1
1
and
2
2
are similar.
Drag statements and reasons to complete the proof.
pls help w imagine math. I do not understand how to do the question
1, 1 and 2, 2 do not define triangles. Cannot drag what is not shown. Cannot copy and paste here.
biggest oof in history of biggest oofs
To prove that the slope between any two points on a straight line is the same, Aaron can use the fact that triangles 1 and 2 are similar. Here are the statements with their reasons to complete the proof:
Statements:
1. Triangle 1 is similar to Triangle 2.
2. Triangle 1 has side lengths a, b, and c, and Triangle 2 has corresponding side lengths ka, kb, and kc (where k is a constant ratio).
3. Let point A(x₁, y₁) and B(x₂, y₂) be two arbitrary points on the straight line.
4. The distance between points A and B is given by √((x₂ - x₁)² + (y₂ - y₁)²).
5. The distance between points A and B in Triangle 1 is given by √(a² + b²).
6. The distance between points A and B in Triangle 2 is given by √((ka)² + (kb)²).
Reasons:
A. Given.
B. Corresponding sides of similar triangles are proportional.
C. Arbitrary points on the straight line.
D. Distance formula between two points in a coordinate system.
E. Definition of the distances in Triangle 1.
F. Definition of the distances in Triangle 2.
By showing that the distances between the two points A and B in both Triangle 1 and Triangle 2 are equal, it can be concluded that the slope between any two points on a straight line is the same.
To prove that the slope between any two points on a straight line is the same, we can use the concept of similar triangles. Here's how you can complete the proof:
1. Given: Triangles 1 and 2 are similar.
- Reason: This is already stated in the question.
2. Let's choose two arbitrary points A and B on the straight line.
- Reason: We want to show that the slope between any two points on the straight line is constant.
3. Connect points A and B to a third point C that lies on the line.
- Reason: By connecting three points, we can form two triangles – triangle ABC and one of the similar triangles (let's say triangle 1).
4. The corresponding sides of triangle ABC and triangle 1 are in proportion.
- Reason: Since triangle ABC and triangle 1 are similar (given), their corresponding sides are in proportion.
5. The corresponding sides AB and AC of triangle ABC correspond to the corresponding sides in triangle 1.
- Reason: We are comparing sides AB and AC in triangle ABC with the corresponding sides in triangle 1.
6. The corresponding sides AB and AC are proportional to the corresponding sides in triangle 1.
- Reason: Based on step 5, we can conclude that the corresponding sides are proportionate.
7. The ratio of the rise (vertical distance) to the run (horizontal distance) in triangle ABC is equal to the ratio in triangle 1.
- Reason: The rise and run for AB in triangle ABC correspond to the rise and run for triangle 1.
8. Therefore, the slope between any two points A and B on the straight line is the same.
- Reason: The ratio of the rise to the run in triangle ABC matches the ratio of the slope in triangle 1, which means that the slope between any two points on the straight line is constant.
By following these steps, you can complete the proof that the slope between any two points on a straight line is the same using the concept of similar triangles.