Diagonal line, a geometric term, is defined as a line segment connecting any two non adjacent vertices of a polygon, or a line segment connecting any two vertices of a polyhedron that are not on the same surface.In additionAlgebraMedium,n-order determinant , the number from top left to bottom right isMajor diagonal, numbers from bottom left to top right are sub diagonal.The word "diagonal" comes from the relationship between "horn" and "horn" in ancient Greek[1], later pulled into Latin ("slash").
connectpolygonOf any two nonadjacent verticesline segmentOr a line segment connecting any two vertices of a polyhedron that are not on the same face
Starting from a vertex of an n-polygon, n-3 diagonal lines can be drawn
N-sided shapes share n × (n-3) ÷ 2 diagonals
◎ Knowledge about rectangular diagonal:
Length × length+width × width=diagonal × diagonal (actuallyPythagorean theorem）That is, the sum of the squares of the two right angled sides is equal to the square of the hypotenuse.
Diagonal line in a narrow sense is the line (line segment) connecting any two non adjacent vertices in a polygon
The generalized diagonal is the line (line segment) connecting any two non adjacent vertices in the multidimensional body
Algebra
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determinant
Figure 1 Determinant diagonal rule
stayn-order determinant In, the number from top left to bottom right is classified as the main diagonal, and the number from bottom left to top right is classified as the sub diagonal.
Cramer's rule: multiply the numbers of the main diagonals respectively and add the values obtained;The numbers of the sub diagonals are multiplied respectively, and the opposite numbers of the obtained values are added.The sum of the two is the value of the determinant.This method is only applicable to determinants of order less than 4.(See Figure 1 on the right)
matrix
A m × n ordermatrixThe diagonal of is the whole of all elements in row k and column k, k=1,2,3... min {m, n}.
aggregate
Let X and Y be any two sets, which are formed by defining all order pairs (x, y):
X×Y := {(x,y)|(x∈X）∧（y∈Y)}
It is called the direct product or Cartesian product of the set X, Y (in order), and X × X is called X^2。
Diagonal lines in sets：
△ = {(a,b）∈X^2| a = b }
Yes X^2In fact, a △ b represents (a, b) ∈△.That is, a=b.
quadrilateral
Illustration
The position, shape and size of a triangle can be determined from its three vertices;When no vertex is given, the shape and size of a triangle can also be determined by some elements of the triangle (six elements in total, three sides and three interior angles of the triangle respectively).Once the triangle is determined, we can study the centerline, height, angular bisectorMedian lineThese are some important line segments.In a quadrilateral, it is studied by dividing it into triangles through diagonals, so the diagonal in the quadrilateral becomes more important.In this paper, how to use the diagonal of a quadrilateral to judge a special quadrilateral is analyzed with examples for reference.[2]
IUsing Diagonal Lines to Judge Special Quadrilateral
In class, we have explored the following important conclusions:
(1) The quadrangle whose diagonals are bisected isparallelogram；
(2) Diagonal lines are equally divided and equalquadrilateralIs a rectangle;
⑶ The diagonals are bisected and perpendicular to each otherdiamond；
⑷ Quadrilateral shapes with equal diagonals and mutually perpendicular bisection aresquare;
In fact, these conclusions are related.As shown in Figure 1, in quadrilateral ABCD, two diagonals intersect at point O.
⑴ When OA=OC, OB=OD, quadrilateral ABCD is a parallelogram.
⑵ When adding AC=BD condition on the basis of OA=OC, OB=OD, quadrilateral ABCDparallelogramBecomes a rectangle on the basis of.
⑶ When the condition AC ∨ BD is added on the basis of OA=OC, OB=OD, the quadrilateral ABCD becomes a diamond on the basis of the parallelogram.
⑷ When the condition AC=BD, AC ∨ BD is added on the basis of OA=OC, OB=OD, the quadrilateral ABCD becomes a square on the basis of the parallelogram.
(5) When AB//CD, and OA=OB, the quadrilateral ABCD is a trapezoid with equal diagonals, that is, isosceles trapezoid.
It can be seen that the transformation of a general quadrilateral into a special quadrilateral can be completed by changing the size relationship and position relationship of the two diagonals.This is also one of the important links between special quadrangles.
IIUsing Diagonal Lines to Determine the Shape of Dynamic Quadrilateral
As shown in Figure 2, point O is a moving point on edge AC, and P is a point on the extension line of BC.Make a straight line MN/BC through point O, set the bisector of MN ∠ BCA at point E, and the bisector of ∠ PCA at point F, connecting AE and AF.
(1) Is there an isosceles triangle in the figure?
⑵ When point O moves to where, is the quadrilateral AECF rectangular?Briefly explain the reasons.
⑶ Can the rectangle in ⑵ be a square?What conditions should be met at this time?
Analysis: ⑴ There is isosceles triangle in Figure 2.
reason:
It's an isosceles triangle.
⑵ When point O moves to the midpoint of AC, the quadrilateral AECF is a rectangle.The reasons are as follows.
From (1).
O is the midpoint of AC.
So:
Therefore, the two diagonal lines AC and EF of the quadrilateral AECF are equally divided and equal to each other.Therefore, quadrilateral AECF is rectangular.
Therefore, when point O moves to the midpoint of AC, the quadrilateral AECF is a rectangle.
⑶ The rectangle in ⑵ may be a square.
Reason: Because of MN/BC, when ∠ ACB=90 °, ∠ AOE=∠ ACB=90 °, that is, diagonal AC and EF are perpendicular to each other.
So the quadrilateral AECF is a square.
That is, when ∠ ACB=90 °, the rectangular AECF in ⑵ is square.
Other non mathematical applications
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1. In engineering, diagonal bracket is used to support the beam of rectangular structure (such as scaffold) to withstand the force pushed into it;Although called diagonal, it is usually not connected to the corners of the rectangle due to practical considerations.
2. Diagonal pliers refer to the steel wire pliers defined by the cutting edge of the knife edge. They intersect with the joint rivet at an angle or form a "diagonal", hence the name.
3. Diagonal binding is a binding type used to combine wing beams or rods, so that the straps cross the rods at a certain angle.
4. In English football, the diagonal control strategy is that the referee and assistant referee position themselves in one of the four quadrants of the court.
