6
The rectangle and its properties
Textbook: pp. 20–22
GoalKnow the definition and properties of a rectangle, including equal diagonals.
New words
rectangle · to‘g‘ri to‘rtburchakequal diagonals · diagonallari tengspecial case · xususiy holconverse theorem · teskari teorema
Explanation
A parallelogram with all angles right is called a rectangle. It is a special case of a parallelogram, so it has all the parallelogram properties: opposite sides are equal and the diagonals bisect each other. Its own property is that the diagonals of a rectangle are equal. The proof uses the right triangles ACD and DBA, which are congruent by their two legs. The converse also holds: a parallelogram with equal diagonals is a rectangle. The point where the diagonals meet is equally far from all four vertices.
Worked examples
A rectangle has perimeter 52 cm and its length is 6 cm more than its width. 2(x + x + 6) = 52, x = 10; the sides are 10 cm and 16 cm.
In rectangle ABCD, AC = 26 cm. The diagonals are equal, so BD = 26 cm; if O is where they meet, OB = 26 : 2 = 13 cm.
Class activity
“Check the corners”: students measure the two diagonals of a rectangular sheet with a string to check that it is a rectangle.
Practice
1
The sides of a rectangle are 7 cm and 12 cm. What is its perimeter (cm)?
38
2
The diagonal BD of a rectangle is 34 cm and the diagonals meet at O. How many cm is AO?
17
3
The diagonals of a parallelogram are 15 cm and 15 cm. What kind of quadrilateral is it?
A rectangle (converse theorem).
4
Why is triangle AOB isosceles in a rectangle?
The diagonals are equal and bisect each other, so AO = BO.