See the module, Construction for details of this. The theorem above suggests three possible tests for a triangle to be isosceles. 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:.
The centre of this rotation symmetry is the circumcentre O described above, because the vertices are equidistant from it. 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. Since the diagonals of a parallelogram bisect each other, a parallelogram has rotation symmetry of order 2 about the intersection of its diagonals. 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.
There are two such lines in a rectangle, so a rectangle has two axes of symmetry meeting right angles. It may seem obvious to the eye that the intersection of these two axes of symmetry is the circumcentre of the rectangle, which is intersection of the two diagonals.
This is illustrated in the diagram to the right, but it needs to be proven. Use the diagram to the right to prove that the line through the midpoints of opposite sides of a rectangle bisects each diagonal. 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.
A quadrilateral whose diagonals bisect each other is a parallelogram, as we will show in this exercise. One of the properties of a parallelogram is that its diagonals bisect each other. This is a converse theorem - that shows that if the diagonals bisect each other, the quadrilateral must be a parallelogram. Show that ABCD is a parallelogram. Mensuration Exponents and Powers Direct and Inverse Proportions Factorisation Introduction to Graphs Q4 Name the quadrilateral whose diagonals: i bisect each other.
Looking to do well in your science exam? Learn from an expert tutor. Book a free class! Solution: i If diagonals of a quadrilateral bisect each other then it is a rhombus, parallelogram, rectangle or square. Set your child up for success with Lido, book a class today!
Maths Class 6 Class 7 Class 8. In concave ones a boomerang for example , they do no intersect. Hope this helped and it's not too late. Mar 18, If, for example, the word bisect is used in a different context, then it only means to divide in two parts. But in Geometry, it means to divide in two equal parts. Word bi means two and word sect means to cut. The answer therefore, as already posted above, is Trapezoid. IHateFactorial New member.
Feb 8,
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