4. Combination problems on square, rectangle and triangle

Combination problems on square, rectangle and triangle

Introduction to square, rectangle

Square is one of the most basic shapes. A square is a quadrilateral with four equal sides & four equal angles.
A rectangle is a quadrilateral with four right angles. In a rectangle, all the angles are equal (360°/4=90°). Furthermore, the opposite sides are parallel and equal, and diagonals bisect each other.

Special case of square

A convex quadrilateral is a square in special conditions as follows:

• A rectangle with two adjacent equal sides.
• A rhombus with a right angle, all angles equal, equal diagonals.
• A quadrilateral with four equal sides and four right angles makes a square.
• A parallelogram with one right angle and two adjacent equal sides makes a square.
• An isosceles trapezoid with equal diagonals, base angle equals.
• A Kite with two disjoint pairs of adjacent sides is equal.

  Area, perimeter & diagonal of square, rectangle, triangle

Diagonals of a square: Diagonals of a square are equal in length, they bisect the angles, and they are the perpendicular bisectors of each other.
Length of the diagonal d=√(a²+a²)=√(2a²2)=a√2 units.
Where d denotes a diagonal of a square is equal to side length times square root of 2.

Diagonals of rectangle: A rectangle has two diagonals they are equal in length and intersect in the middle. The diagonal is the square root of (width squared + height squared).

Diagonal(d)= l²+b²−−−−−−√
Where l is the length of the rectangle.

Where b is the breadth of the rectangle.
Perimeter is the actual distance around a closed figure.
2. Perimeter of a regular polygon = Number of sides x Length of one side
3. Perimeter of a square = 4 x side

 

Perimeter of a triangle = AB + BC + CA (Sum of all sides of triangle)

Perimeter of a rectangle = 2 [length + breadth]

= 2(l+ b)

Types of triangle
Right angle triangle: A triangle where one of its interior angles is a right angle 90°.

Area:

Area(A)=1/2(b×h)

Thus, the height of the triangle h=Area×2/b

And, the base of triangle b=Area×2/h

Where h is denoted as height.

Where b is denoted as base.

The perimeter:

a²+b²=c²

a, b  are the lengths of the other two sides.

Where c is the length of the hypotenuse.

 Sides: The two sides that are not the hypotenuse makes the right angle.

Hypotenuse:  The side opposite the right angle is called the hypotenuse. It will always be the longest side of a right triangle. 

Isosceles triangle: A triangle which has two of its sides equal in length.

Area:

Area(A)=1/2(b×h)

Thus, the height of the triangle h=Area×2/b

And, the base of the triangle b=Area×2/h

Where h is denoted as height.

Where b is denoted as base.

Altitude h=√(a²−b²)/4

The perimeter:

P=2a+b

Where a is the lengths of the two equal sides.

Where b is the lengths of the other sides.

Equilateral triangle: A triangle which has all three of its sides equal in length.

Area:

Area(A)=√3/4s².

Where s² denotes sides of the triangle.

 The perimeter:

Perimeter(P)=a+b+c or P=s+s+s.

a, b, c are the lengths of the three equal sides.

or

S is the lengths of the three equal sides.