Need help? Tutors are standing by. Ask a question or Get instant tutoring Need help? Tutors are available. Find a geometry tutor. What you learned: 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.
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The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future. There are different properties of parallelograms. Opposite sides are equal. Opposite angle are equal Diagonals bisect each other. A diagonal of a parallelogram divides it into two congruent triangles. Some Special Parallelograms Depending on the properties, there are three special types of parallelogram: Rectangle Rhombus Square Rectangle A rectangle is a special type of parallelogram which has all the properties of parallelogram along with some different properties.
Each angle of a rectangle must be a right angle, i. Properties of a Rectangle Properties of a rectangle are similar to those of a parallelogram: Opposite Sides are parallel to each other. Opposite Sides of a rectangle are equal. Diagonals bisect each other Diagonals of the rectangle are equal.
Each interior angle of a rectangle is equal i. So all rectangles are parallelogram but all parallelograms are not rectangle.
Find the length of diagonal BD. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials.
Quadrilaterals: Classification A quadrilateral is a polygon with four sides. There are many special types of quadrilateral. A parallelogram also has the following properties: Opposite angles are congruent; Opposite sides are congruent; Adjacent angles are supplementary; The diagonals bisect each other. A rectangle has all the properties of a parallelogram, plus the following: The diagonals are congruent.
A rhombus has all the properties of a parallelogram, plus the following: The diagonals intersect at right angles. 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. Trapezia also have a characteristic property involving the diagonals, but the property concerns areas, not lengths or angles.
A trapezium is a quadrilateral with one pair of opposite sides parallel. Using co-interior angles, we can see that a trapezium has two pairs of adjacent supplementary angles. Conversely, if a quadrilateral is known to have one pair of adjacent supplementary angles, then it is a trapezium. The diagonals of a convex quadrilateral dissect the quadrilateral into four triangular regions, as shown in the diagrams below.
In a trapezium, two of these triangles have the same area, and the converse of this property is a test for a quadrilateral to be a trapezium. These results are written as exercises because they are not usually regarded as standard theorems for students to know. The trapezia that occur in this exercise are called isosceles trapezia.
This module completes the study of special quadrilaterals using congruence. Similarity is a generalisation of congruence, and when it has been developed, some further results about special quadrilaterals will become possible. All triangles have both a circumcircle and an incircle. The only quadrilaterals that have a circumcircle are those with opposite angles supplementary, the situation with incircles is interesting.
For example, a rhombus always has an incircle. As an easy exercise show that if the lengths of the diagonals of the rhombus are p and q and the radius of the incircle is r then. If the lengths of the diagonals are p and q show that :. When complex numbers are graphed on Argand diagrams, many arithmetic and algebraic results are proved or illustrated using special quadrilaterals. This illustrates very well the constant attitude in mathematics that an investigation is not complete until a theorem with a true converse has been identified.
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