A square is always a rhombus, whereas each of a parallelogram, a trapezoid, and a rectangle do not have to be. A rhombus is a quadrilateral with all four sides being of equal length. To see which of the given shapes fall into this category, let's look at their definitions.
A parallelogram is a quadrilateral with two pairs of parallel sides. A trapezoid is a quadrilateral with at least one pair of parallel sides. A rectangle is a quadrilateral with four right angles.
Use the other two objects to connect the original two, up and to the right, to make your four-sided quadrilateral , plane figure --a rhombus. Look at the bottom left angle and the top right angle. They are the same. They are congruent. Look at the bottom right angle and the top left angle: they are congruent. Opposite interior angles of a rhombus are congruent. A wonderful and rare property of a rhombus is that its diagonals are always perpendicular to each other.
You can see this for yourself if you lay down your four straight objects to make a rhombus and then draw in diagonals. No matter what angles you have for the rhombus's four vertices, the diagonals of a rhombus are always at right angles to each other.
These diagonals also cut each other exactly in half. Geometricians say they bisect each other. That means the two diagonals divide the rhombus up into four right-angle triangles. Once you watch this lesson and read about a rhombus, you will know how this plane figure fits into the whole family of plane figures, what properties make a rhombus unique, and how to recognize a rhombus by finding its two special identifying properties. How To Find the Area of a Rhombus.
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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.
Notice that the radii of the arcs meeting at P need not be the same as the radius of the first arc with centre O. Notice that the radii of the arcs meeting at Q need not be the same as the radii of the original arc with centre P. In the diagram to the left, the radii of the arcs meeting at P are not the same as the radii of the arcs meeting at Q. Trapezoid A trapezoid is when there is only one pair of opposing parallel sides see the picture below.
For more information, see Kids Math: Quadrilaterals. I found an answer from www. May 15, For more information, see Why is a square always a rhombus , but a rhombus is not always a The rhombus has a square as a special case, and is a special case of a kite and parallelogram. In plane Euclidean geometry, a rhombus plural rhombi or rhombuses is a quadrilateral whose Not every parallelogram is a rhombus , though any parallelogram with perpendicular diagonals the second property is a rhombus For more information, see Rhombus - Wikipedia.
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