prove that a rhombus is a parllelogram.
Answers
Answer:
All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram. ... The same can be done for the other two sides, and know we know that opposite sides are parallel. Therefore, a rhombus is a parallelogram.
Answer:
Whether a parallelogram is a rhombus, here are their comparative properties.
Parallelogram. Its properties are
(a) Opposite sides are equal and parallel.
(b) Opposite angles are equal.
(c) Diagonals bisect each other.
(d) The diagonals bisect the parallelogram into two congruent triangles.
(e) Any two adjacent angles add up to 180 degrees.
(f) The angle bisectors of the opposite angles of a parallelogram are parallel.
(g) The angles bisectors of two adjacent angles form a right angle where they meet.
(h) ) The angle bisectors of all the 4 angles form a rectangle inside the parallelogram.
(i) The sum of the four exterior angles is 4 right angles.
(j) The sum of the four interior angles is 4 right angles.
(k) Join the midpoints of the four sides in order and you get another parallelogram.
Rhombus. Its properties are
(a) All sides are equal.
(b) Opposite sides are parallel.
(c) Opposite angles are equal.
(d) Diagonals bisect each other at right angles.
(e) Diagonals bisect the angles.
(f) Any two adjacent angles add up to 180 degrees.
(g) The sum of the four exterior angles is 4 right angles.
(h) The sum of the four interior angles is 4 right angles.
(i) The two diagonals form four congruent right angled triangles.
(j) Join the mid-points of the sides in order and you get a rectangle.
(k) Join the mid-points of the half the diagonals in order and you get a rhombus.
(l) The distance of the point of intersection of the two diagonals to the mid point of the sides will be the radius of the circumscribing of each of the 4 right-angled triangles.
(m) The area of the rhombus is a product of the lengths of the 2 diagonals divided by 2.
(n) The lines joining the midpoints of the 4 sides in order, will form a rectangle whose length and width will be half that of the main diagonals. The area of this rectangle will be one-fourth that of the rhombus.
(o) If through the point of intersection of the two diagonals you draw lines parallel to the sides, you get 4 congruent rhombus each of whose area will be one-fourth that of the original rhombus.
(p) There can be no circumscribing circle around a rhombus.
(q) There can be no inscribed circle within a rhombus.
(r) Two congruent equilateral triangles are formed if the shorter diagonal is equal to one of the sides.
(s) Two congruent isosceles acute triangles are formed when cut along the shorter diagonal.
(t) Two congruent isosceles obtuse triangles are formed when cut along the longer diagonal.
(u) Four congruent RATs are formed when cut along both the diagonals. These RATs cannot be isosceles RATs.
(v) Join the quarter points of both the diagonals and you get a similar rhombus of 1/4th area as the parent rhombus.
(w) Rotate the rhombus around the major diagonal, you get a double cone (cone having the same base) whose total height is more than the common base diameter.
(x) Rotate the rhombus around the minor diagonal, you get a double cone (cone having the same base) whose total height is less than the common base diameter