"Special Parallelograms" Quadrilateral PPT courseware

Description

"Special Parallelograms" Quadrilateral PPT courseware

Definition of rectangle

A parallelogram with one right angle is called a rectangle, that is, a rectangle.

A rectangle is a parallelogram with one internal angle that is a right angle. What are the characteristics of the other internal angles? What are the characteristics of two diagonals?

Properties of rectangle:

The four corners of a rectangle are right angles;

The diagonals of a rectangle are equal.

The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse

Explanation: In the rectangle ABCD, suppose the diagonals AC and BD intersect at point O, then AC=BD. According to the properties of parallelograms, the diagonals bisect each other, that is, OA=OC, OB=OD. So OA=OC =OB=OD=1/2AC=1/2BD.

Example questions

Example 1 As shown in the figure, the two diagonals of the rectangle ABCD intersect at point O, ∠AOB=60°, AB=4cm, find the length of the diagonal of the rectangle.

Solution: Since quadrilateral ABCD is a rectangle,

Therefore AC and BD are equal and bisect each other.

So OA=OB.

And AOB=600.

So △OAB is an equilateral triangle.

OA=AB=4cm.

Therefore, the diagonal length of the rectangle is AC=BD=2OA=8cm.

think

It can be seen from the definition of a rectangle that a parallelogram with one right angle is a rectangle. When one angle of the parallelogram becomes a right angle, the other three angles become right angles, and the two diagonals also become equal line segments.

Is there any other way to turn a parallelogram into a rectangle?

practice

1. It is known that the perimeter of the rectangle is 24, and the ratio of two adjacent sides is 1:2, then the area of ​​the rectangle is ____________

2. The angle between the two diagonals of a rectangle is 60°, and the length of one side is 10, then the length of the other side is ____________

3. Please write your conclusion on the horizontal line and fill in the reasons in the brackets.

∵ Quadrilateral ABCD is a rectangle

∴__________ ( )

Class exercises

True or False

1. A quadrilateral with equal diagonals and equal opposite sides is a rectangle. ( )

2. A quadrilateral with equal distances from the intersection of its two diagonals to its four vertices is a rectangle. ( )

3. A quadrilateral with equal opposite sides and right angles is a rectangle. ( )

4. A quadrilateral with three equal angles is a rectangle. ( )

Multiple choice questions

5. A quadrilateral that meets the conditions ____ is a rectangle. ( )

A. The two diagonals are equal B. Diagonals are perpendicular to each other

C. A set of opposite angles is a right angle D. There are three right angles

6. The condition that can determine whether a quadrilateral is a rectangle is ( )

A. Diagonals are equal B. diagonal vertical

C. The diagonals bisect each other and are equal D. Diagonals are vertical and equal

summary

1. What is a rectangle?

2. What are the properties of a rectangle?

3. How to determine whether a parallelogram is a rectangle?

4. How to determine whether a quadrilateral is a rectangle?

5. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.

induction

A rhombus is an axially symmetrical figure, and the straight line where its diagonal lies is its axis of symmetry.

According to the symmetry of the rhombus, the following properties of the rhombus can be easily obtained:

All four sides of a rhombus are equal;

The two diagonals of a rhombus are perpendicular to each other, and each diagonal bisects a set of opposite angles.

practise

1. Quadrilateral ABCD is a rhombus, point O is the intersection of the two diagonals, AB=5cm, AO=4cm, find the lengths of the two diagonals AC and BD.

2. The lengths of the two diagonals of the rhombus are 6cm and 8cm respectively. Find the perimeter and area of ​​the rhombus.

3. It is known that the circumference of the rhombus is 12cm, then its side length is ______.

4. In rhombus ABCD, ∠ABC=60 degrees, then ∠BAC=_______.

5. The lengths of the two diagonals of the rhombus are 6cm and 8cm respectively, then the area of ​​the rhombus is _____.

think

What are the properties of a square? How to tell if a quadrilateral is a square? Write them out, share them with classmates, and then prove some of the conclusions? .

Prove: The two diagonals of a square divide the square into four congruent isosceles right triangles.

Known: As shown in the figure, quadrilateral ABCD is a square, and diagonals AC and BD intersect at point O.

Prove: ABO, BCO, CDO, and DAO are congruent isosceles right triangles

Proof: Since quadrilateral ABCD is a square,

So AC=BD, AC⊥BD.

AO=BO=CO=DO.

Therefore, △ABO, △BCO, △CDO, and △DAO are all isosceles right triangles, and △ABO≌△BCO≌△CDO≌△DAO.

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Update Time: 2024-10-02

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