From each corner of a parallelogram a perpendicular is drawn upon the diagonal which
does not pass through that corner and these are produced to form another parallelogram,
show that its diagonals are perpendicular to the sides of the first parallelogram
and that they both have the same centre.
To show that the diagonals of the second parallelogram are perpendicular to the sides of the first parallelogram, as well as the fact that they share the same center, we can use the concept of vectors.
Let's assume that the given parallelogram has vertices A, B, C, and D, and its diagonals intersect at point O.
First, we will find the equation for the diagonals of the given parallelogram:
1. Find the midpoint of the diagonal AC:
- The midpoint of AC can be found using the formula:
Midpoint of AC = (A + C) / 2
2. Determine the direction vector for the diagonal AC:
- The direction vector of AC can be found by subtracting the coordinates of A from C:
Direction vector AC = C - A
3. Write the equation for the diagonal AC:
- The equation of the line passing through the midpoint of AC with the direction vector AC is:
AC: r = (A + C)/2 + t * (C - A)
Similarly, we can find the equation for the diagonal BD:
4. Find the midpoint of the diagonal BD:
- The midpoint of BD can be found using the formula:
Midpoint of BD = (B + D) / 2
5. Determine the direction vector for the diagonal BD:
- The direction vector of BD can be found by subtracting the coordinates of B from D:
Direction vector BD = D - B
6. Write the equation for the diagonal BD:
- The equation of the line passing through the midpoint of BD with the direction vector BD is:
BD: r = (B + D)/2 + s * (D - B)
Now, we can prove that the diagonals are perpendicular to the sides of the first parallelogram:
7. Show that the direction vectors of the diagonals are perpendicular to the sides of the parallelogram:
- Since we have the direction vectors of AC and BD, let's take the dot product of these vectors with the sides of the parallelogram, AB and BC.
- If the dot product of the direction vector and the side vector is zero, it means that the vectors are perpendicular.
- Calculate the dot product of AC and AB, and AC and BC, and verify that they are zero.
Finally, we can prove that the diagonals share the same center:
8. Show that the diagonals intersect at the same point:
- We already have the equations for the diagonals, AC and BD.
- Solve these equations simultaneously to find the point of intersection, which should be the same point O.
By following these steps, you can demonstrate that the diagonals of the second parallelogram are perpendicular to the sides of the first parallelogram and that they share the same center.
To show that the diagonals of the second parallelogram are perpendicular to the sides of the first parallelogram and that they both have the same center, we can follow these steps:
Step 1: Draw a diagram
Start by drawing a parallelogram. Label its corners as A, B, C, and D.
Step 2: Identify the diagonals
In the first parallelogram, join the opposite corners by drawing the diagonals. Label the point where the diagonals intersect as O.
Step 3: Draw perpendiculars
From each corner of the parallelogram (A, B, C, and D), draw a perpendicular line to the diagonal that does not pass through that corner. Extend these perpendicular lines until they intersect and form another parallelogram.
Step 4: Label the new parallelogram
In the second parallelogram, label the new corners as P, Q, R, and S. The diagonal PR is formed by extending the perpendicular from corner A, and the diagonal QS is formed by extending the perpendicular from corner B.
Step 5: Prove diagonals are perpendicular
To show that the diagonals PR and QS are perpendicular to the sides of the first parallelogram, we need to prove that the angles they form are right angles.
Step 6: Use properties of parallel lines
Since the second parallelogram is formed by extending perpendiculars from the corners of the first parallelogram, the sides of the second parallelogram are parallel to the corresponding sides of the first parallelogram.
Step 7: Prove angles are right angles
Since opposite sides of a parallelogram are parallel, and the perpendicular lines are drawn from the corners, the angles formed by the diagonals PR and QS with the sides of the first parallelogram are 90 degrees (right angles).
Step 8: Diagonals have the same center
To show that both parallelograms have the same center, we need to prove that the point of intersection O of the diagonals of the first parallelogram is also the point of intersection of the diagonals PR and QS of the second parallelogram.
Step 9: Use symmetry and congruence
Since the second parallelogram is formed by extending perpendiculars from the corners of the first parallelogram, and the diagonals of the first parallelogram intersect at point O, the diagonals PR and QS of the second parallelogram must also intersect at point O. This shows that both parallelograms have the same center.
By following these steps, you can demonstrate that the diagonals of the second parallelogram are indeed perpendicular to the sides of the first parallelogram, and that they both have the same center.