Just like rectangles are a special type of parallelogram , squares are a special type of rectangles, in which all the sides are equal. Or you can think of it as a special type of rhombus diamond in which all the angles are right angles.
Because all squares are also both rectangles and diamonds, they combine all the properties of both diamonds and rectangles. In a rectangle, the diagonals are equal and bisect each other. And in a diamond, the diagonals are perpendicular to each other. 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.
A rhombus is a quadrilateral with all sides equal. A rhombus thus has all the properties of a parallelogram:. When drawing a rhombus, there are two helpful orientations that we can use, as illustrated below. 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. It is very straightforward to construct a rhombus using the definition of a rhombus. The figure OAPB is a rhombus because all its sides are 5cm. Use the cosine rule or drop a perpendicular and use simple trigonometry to find the lengths of the lengths of the diagonals of the rhombus OAPB constructed above. This leads to yet another way to construct a line parallel to a given line through a given point P.
The exercise above showed that each diagonal of a rhombus dissects the rhombus into two congruent triangles that are reflections of each other in the diagonal,. Thus the diagonals of a rhombus are axes of symmetry.
The following property shows that these two axes are perpendicular. The proof given here uses the theorem about the axis of symmetry of an isosceles triangle proven at the start of this module.
Two other proofs are outlined as exercises. The diagonals of a rhombus are perpendicular. The diagonals also bisect each other because a rhombus is a parallelogram, so we usually state the property as. We now turn to tests for a quadrilateral to be a rhombus. This is a matter of establishing that a property, or a combination of properties, gives us enough information for us to conclude that such a quadrilateral is a rhombus.
We have proved that the opposite sides of a parallelogram are equal, so if two adjacent sides are equal, then all four sides are equal and it is a rhombus. A quadrilateral whose diagonals bisect each other at right angles is a rhombus. It follows similarly that. A quadrilateral whose diagonals bisect each other is a parallelogram, so this test is often stated as.
This figure is a rhombus because its diagonals bisect each other at right angles. If the circles in the constructions above have radius 4cm and 6cm, what will the side length and the vertex angles of the resulting rhombus be? If each diagonal of a quadrilateral bisects the vertex angles through which it passes, then the quadrilateral is a rhombus. Let ABCD be a quadrilateral, and suppose the diagonals bisect the angles, then let.
The converse of a property is not necessarily a test. The following exercise gives an interesting characterisation of quadrilaterals with perpendicular diagonals. One half is straightforward, the other requires proof by contradiction and an ingenious construction. We usually think of a square as a quadrilateral with all sides equal and all angles right angles. Now that we have dealt with the rectangle and the rhombus, we can define a square concisely as:.
A square thus has all the properties of a rectangle, and all the properties of a rhombus. The intersection of the two diagonals is the circumcentre of the circumcircle through all four vertices.
We have already seen, in the discussion of the symmetries of a rectangle, that all four axes of symmetry meet at the circumcentre. A square ABCD is congruent to itself in three other orientations,. The centre of the rotation symmetry is the circumcentre, because the vertices are equidistant from it. The most obvious way to construct a square of side length 6cm is to construct a right angle, cut off lengths of 6cm on both arms with a single arc, and then complete the parallelogram.
Alternatively, we can combine the previous diagonal constructions of the rectangle of the rhombus. Construct two perpendicular lines intersecting at O , draw a circle with centre O , and join up the four points where the circle cuts the lines. What radius should the circle have for the second construction above to produce a square of side length 6cm? Some of the distinctive properties of the diagonals of a rhombus hold also in a kite, which is a more general figure.
Because of this, several important constructions are better understood in terms of kites than in terms of rhombuses. A kite is a quadrilateral with two pairs of adjacent equal sides.
A kite may be convex or non-convex, as shown in the diagrams above. The definition allows a straightforward construction using compasses. The last two circles meet at two points P and P 0 , one inside the large circle and one outside, giving a convex kite and a non-convex kite meeting the specifications. Notice that the reflex angle of a non-convex kite is formed between the two shorter sides. What will the vertex angles and the lengths of the diagonals be in the kites constructed above?
The congruence follows from the definition, and the other parts follow from the congruence. Using the theorem about the axis of symmetry of an isosceles triangle, the bisector AM of the apex angle of the isosceles triangle ABD is also the perpendicular bisector of its base BD. The converses of some these properties of a kite are tests for a quadrilateral to be a kite. If one diagonal of a quadrilateral bisects the two vertex angles through which it passes, then the quadrilateral is a kite.
If one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. Is it true that if a quadrilateral has a pair of opposite angles equal and a pair of adjacent sides equal, then it is a kite?
Three of the most common ruler-and-compasses constructions can be explained in terms of kites. Its diagonals divide the figure into 4 congruent triangles. Its diagonals are perpendicular bisectors of eachother. In other words, if the parallelogram is a rhombus. The intersection of the diagonals of a kite form 90 degree right angles.
This means that they are perpendicular. The longer diagonal of a kite bisects the shorter one. This means that the longer diagonal cuts the shorter one in half. A kite is a quadrilateral with two distinct pairs of congruent adjacent sides.
You can prove Theorem Kite properties include 1 two pairs of consecutive, congruent sides, 2 congruent non-vertex angles and 3 perpendicular diagonals. Other important polygon properties to be familiar with include trapezoid properties, parallelogram properties, rhombus properties, and rectangle and square properties. In Euclidean geometry, a right kite is a kite a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other that can be inscribed in a circle.
Thus the right kite is a convex quadrilateral and has two opposite right angles. If the two pairs of opposite sides of a quadrilateral have equal sums of squares, then the diagonals of the quadrilateral are perpendicular to each other. A rhombus has all sides equal, while a rectangle has all angles equal. A rhombus has opposite angles equal, while a rectangle has opposite sides equal. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
All of the properties of a parallelogram apply the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary. The diagonals bisect the angles. The diagonals are perpendicular bisectors of each other.
The diagonals of a trapezoid are perpendicular. An equilateral parallelogram is equiangular. The angles of a parallelogram are congruent. If a quadrilateral has three angles of equal measure, then the fourth angle must be a right angle. A trapezium or a trapezoid is a quadrilateral with a pair of parallel sides. Its diagonals bisect with each other. Now, for the diagonals to bisect each other at right angles, i.
Hence, the diagonals of a parallelogram bisect each other but not necessarily at right angles.
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