Question

Sania saw a sign board of
length, [(√3 + √2)^2 - 2√6] cm
and breadth, [(√3 - √2)^2 + 2√6]
cm. Find the length of the
diagonal of the sign board and
represent it on the number line.​

Answer 1

Step-by-step explanation:

Given :-

Sania saw a sign board of length

[(√3 + √2)^2 - 2√6] cm and breadth

[(√3 - √2)^2 + 2√6]cm.

To find :-

Find the length of the diagonal of the sign board and represent it on the number line ?

Solution :-

Given that

Sania saw a sign board

Length of the board (l) = [(√3+√2)²-2√6] cm

=> l = [(√3)²+2(√3)(√2)+(√2)²-2√6] cm

Since (a+b)² = a²+2ab+b²

Where a = √3 and b=√2

=> l = (3+2√6+2-2√6) cm

=> l = (3+2) cm

=> l = 5 cm

Breadth of the board = [(√3-√2)²+2√6] cm

=> b = [(√3)²-2(√3)(√2)+(√2)²+2√6] cm

Since (a-b)² = a²-2ab+b²

Where a = √3 and b=√2

=> b = (3-2√6+2+2√6) cm

=> b = (3+2) cm

=> b = 5 cm

We know that

If the length and breadth of a rectangle are l units and b units then the length of it's diagonal is

d = √(l²+b²) units

On Substituting these values in the above formula then

=> Length of the diagonal of the board

=> d = √(5²+5²) cm

=> d = √(25+25) cm

=> d =√50 cm or

=> d = √(5×5×2) cm

=> d = 5√2 cm

Answer:-

The length of the diagonal of the sign bord is √50 cm or 5√2 cm

Representing √50 on the number line:-

• Draw a number line and construct OABC rectangle whose length and breadth 5 cm each.
• OA = BC = AB = OC = 5 cm
• join O and B
• OB is the diagonal
• ∆OAB is a right angled triangle
• By Pythagoras theorem, OB² = OA²+AB²
• OB² = 5²+5²
• OB² = 25+25
• OB² = 50
• OB = √50
• Draw an arc of the length √50 units and it cuts the number line at k
• K represents √50 on the number line.

Or

• We use to represent √50 on the number line with Successive Magnification method
• √50 =5√2 = 5×1.414=7.07...

Note :-

The length and breadth are equal in the given sign board so it is a square shaped board

So ,Length of the diagonal =a√2 units

=> d = 5√2 cm

Used formulae:-

• If the length and breadth of a rectangle are l units and b units then the length of it's diagonal is d = √(l²+b²) units
• Length of the diagonal =a√2 units where a is the side of the square.

Pythagoras theorem:-

• In a right angled triangle, The square of the hypotenuse is equal to the sum of the squares of the other two sides.

Used Identities:-

• (a+b)² = a²+2ab+b²
• (a-b)² = a²-2ab+b