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Prove that if the diagonals of a parallelogram bisect each other at right angle then it's a rhombus.
Answers
Answer:
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Answer-Theorem
A quadrilateral whose diagonals bisect each other at right angles is a rhombus.
Proof
Let ABCD be a quadrilateral whose diagonals bisect
each other at right angles at M.
We prove that DA = AB. It follows similarly that
AB = BC and BC = CD
triangleAMB equiv triangleAMD (SAS)
So AB = AD and by the first test above ABCD is a rhombus.
A quadrilateral whose diagonals bisect each other is a parallelogram, so this test is often stated as
‘If the diagonals of a parallelogram are perpendicular, then it is a rhombus.’
Hope this is helpful..
RHOMBUSES
The Greeks took the word rhombos from the shape of a piece of wood that was whirled about the head like a bullroarer in religious ceremonies. This derivation does not imply a definition, unlike the words ‘parallelogram’ and ‘rectangle’, but we shall take their classical definition of the rhombus as our definition because it is the one most usually adopted by modern authors.
Definition of a rhombus
A rhombus is a quadrilateral with all sides equal.
First property of a rhombus − A rhombus is a parallelogram
Since its opposite sides are equal, a rhombus is a parallelogram − this was our second test for a parallelogram in the previous module. A rhombus thus has all the properties of a parallelogram:
Its opposite sides are parallel.
Its opposite angles are equal.
Its diagonals bisect each other.
It has rotation symmetry of order two about the intersection of its diagonals.
When drawing a rhombus, there are two helpful orientations that we can use, as illustrated below.
The rhombus on the left looks like a ‘pushed-over square’, and has the orientation we usually use for a parallelogram. The rhombus on the right has been rotated so that it looks like the diamond in a pack of cards. It is often useful to think of this as the standard shape of a rhombus.
Constructing a rhombus using the definition
It is very straightforward to construct a rhombus using the definition of a rhombus. Suppose that we want to construct a rhombus with side lengths 5cm and acute vertex angle 50°.
Draw a circle with radius 5cm.
Draw two radii OA and OB meeting at 50°
at the centre O.
Draw arcs with the same radius 5cm and centres
A and B, and let P be their point of intersection.
The figure OAPB is a rhombus because all its sides are 5cm.
The symmetries of an equilateral triangle
An equilateral triangle is an isosceles triangle in three different ways, so the three vertex angle bisectors form three axes of symmetry meeting each other at 60°. In an equilateral triangle, each vertex angle bisector is the perpendicular bisector of the opposite side − we proved in the previous module that in any triangle, these three perpendicular bisectors are concurrent. They meet at a point which is the centre of a circle through all three vertices. The point is called the circumcentre and the circle is called the circumcircle of the triangle.
An equilateral triangle is also congruent to itself in two other orientations:
triangleABC equiv triangleBCA equiv triangleCAB (SSS),
corresponding to the fact that it has rotation symmetry of order 3. The centre of this rotation symmetry is the circumcentre O described above, because the vertices are equidistant from it.
Other triangles do not have reflection or rotation symmetry
In a non-trivial rotation symmetry, one side of a triangle is mapped to a second side,
and the second side mapped to the third side, so the triangle must be equilateral.
In a reflection symmetry, two sides are swapped, so the triangle must be isosceles.
Thus a triangle that is not isosceles has neither reflection nor rotation symmetry. Such a triangle is called scalene.
Rotation symmetry of a parallelogram
Since the diagonals of a parallelogram bisect each other, a parallelogram has rotation symmetry of order 2 about the intersection of its diagonals. Joining the diagonal AC of a parallelogram ABCD produces two congruent triangles,
triangleABC equiv triangleCDA (AAS);
Reflection symmetry of a rectangle
A rectangle is a parallelogram, so it has rotationsymmetry of order 2 about the intersection of its diagonals. This is even clearer in a rectangle than in a general parallelogram because the diagonals have equal length, so their intersection is the circumcentre of the circumcircle passing through all four vertices.
The line through the midpoints of two opposite sides of a rectangle dissects the rectangle into two rectangles that are congruent to each other, and are in fact reflections of each other in the constructed line.