show that the quadrilateral formed by joining the mid -points of the sides of a square is also a square.
Use rules for congruent triangles (the four corners that get lopped off) to show that the four sides of the new quadrilateral are equal, and that the corner angles are right angles. That proves it to be a square also.
To prove that the quadrilateral formed by joining the midpoints of the sides of a square is also a square, we need to demonstrate that both pairs of opposite sides are parallel and congruent, and that all angles are right angles.
Let's consider a square with vertices labeled as A, B, C, and D in a counterclockwise direction, and let E, F, G, and H be the midpoints of the sides AB, BC, CD, and DA, respectively.
To establish that the sides of the quadrilateral are parallel and congruent, we can show that the length of any two consecutive sides are equal. By construction, we know that E is the midpoint of AB, so AE = EB. Similarly, F is the midpoint of BC, so BF = FC. Continuing this pattern, CG = GD and DH = HA.
Now, let's examine the remaining two pairs of opposite sides. We can observe that GH is parallel to EF since each side of the square ABGH is parallel to its corresponding side CDHE. Additionally, HE is parallel to FG because each side of the square ABGH is parallel to its corresponding side BCFG.
To verify that all angles are right angles, we can show that any one of the angles is 90 degrees. For instance, angle GHE is a right angle because it is formed by the intersection of the diagonals EG and FH, which are both diagonals of the square ABCD.
Therefore, we have demonstrated that the quadrilateral formed by joining the midpoints of the sides of a square has congruent and parallel sides, as well as right angles. Hence, it can be concluded that this quadrilateral is also a square.