ABCD is a rhombus and AB is produced to E and F such that AE=AB=BF. Prove that ED and FC are perpendicular to each other.
To prove that ED and FC are perpendicular to each other, we need to show that the slopes of the lines ED and FC are negative reciprocals of each other. Here's how you can prove it:
1. Given: ABCD is a rhombus, and AB is produced to E and F such that AE = AB = BF.
2. Draw the diagram: Start by drawing a rhombus ABCD. Extend the line segment AB to points E and F such that AE = AB = BF.
3. Identify the properties of a rhombus: In a rhombus, opposite sides are parallel, opposite angles are equal, and diagonals bisect each other at right angles.
4. Determine the slopes: To prove that ED and FC are perpendicular, we need to show that their slopes are negative reciprocals of each other. Let's calculate the slopes of ED and FC.
- ED: The line ED is the extension of side AD. The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1).
In the case of ED, let's take two points as A(x1, y1) and D(x2, y2).
The coordinates of A and D can be determined by considering the properties of a rhombus.
Since AD is a diagonal, it bisects the angles of the rhombus, so AD is perpendicular to both AB and BC.
Therefore, the slope of AD is equal to the negative reciprocal of the slope of AB.
- FC: The line FC is the extension of side BC. Similar to ED, let's take two points as B(x1, y1) and C(x2, y2).
Since BC is also a diagonal of the rhombus, it is perpendicular to both DC and AD.
Therefore, the slope of FC is equal to the negative reciprocal of the slope of DC.
5. Compare the slopes: Now, compare the slopes of ED and FC. Since both slopes are the negative reciprocals of the slopes of AB and DC, respectively, they will also be negative reciprocals of each other.
6. Conclude: As the slopes of ED and FC are negative reciprocals of each other, we can conclude that ED and FC are perpendicular to each other.
Therefore, we have proven that ED and FC are perpendicular lines based on the given conditions of the rhombus.