Diagonal Formula

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A rectangle has two diagonals. Diagonals are line segments which link the two vertices i.e. corners of the rectangle. The diagonals of the rectangle have the following properties:

1.      The two diagonals are of same length i.e. diagonals are congruent.

2.      Each diagonal bisects the other i.e. the point where they intersect each other divides the diagonal into two equal parts.

3.      Each diagonal divides the rectangle into two congruent triangles. In other words, a diagonal divides a rectangle into triangles of same area. Therefore, each triangle has half of the area of rectangle.

FORMULA

The diagonal divides the rectangle into two equal triangles. Thus, a diagonal is the hypotenuse of these triangles. To find the hypotenuse, we use the Pythagoras theorem if we know the length and width of the rectangle.

Therefore, Diagonal =√(w2 + h2)

                                           Where, w = width of the rectangle

                                                         h = height of the rectangle


 
Problem 1: Find the diagonal of the rectangle with height of 4cms and width of 3cms.

Solution: The diagonal can be calculated by the following steps:

=> Step 1:- Draw a rectangle with height of 6cms and width of 4cms.

=> Step 2:- Draw a diagonal joining the two corners of the rectangle which divides it into two triangles.

=> Step 3:- Now, let a= 4 and b=3. Use the Pythagoras theorem to calculate the length of the diagonal.

=> Pythagoras theorem is as follows:   a^2 + b^2 = c^2, here c is the length of the diagonal

=> Step 4:- Substitute the values in the theorem in the following way = 4^2+ 3^2 = c^2

=> 16 + 9 = c^2 => 25 = c^2

=> 25- c^2 = 0

=> Step 5:- Use the difference of squares to factor this equation => (5 – c) (5 + c) = 0

=> 5 = c

=> Thus, the length of the diagonal is 5cms.



Problem 2: Calculate the diagonal of the rectangle with height of 6cms and width of 4cms.

Solution: The length of the diagonal can be calculated by the formula:

=> Diagonal length = √(w2 + h2) = √(62 + 42)

= √(36 + 16) = 7.2 cms        
 
=> Thus, length of the diagonal is 7.2 cms.

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