Question 16 - End-of-Chapter Exercises - Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)

Transcript

Question 16 Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral. The Square Root Spiral is a beautiful geometric trick for drawing irrational lengths using only a ruler and the Pythagorean theorem (a2 + b2 = c2) Let's calculate hypotenuses one by one to find the hidden patter Triangle 1 (The starting pink triangle) The base is 1 , and the height is 1 . " Hypotenuse "=√(𝟏^𝟐+𝟏^𝟐 )=√(1+1)=√𝟐 Triangle 2 (The purple triangle) The base is the hypotenuse of the last triangle (√2), and the height is 1 . " Hypotenuse "=√((√𝟐 )^𝟐+𝟏^𝟐 )=√(2+1)=√𝟑 Triangle 3 (The blue triangle) The base is now √3, and the height is 1 . " Hypotenuse "=√((√𝟑 )^𝟐+𝟏^𝟐 )=√(3+1)=√𝟒 " = 2" Triangle 4 (The green triangle) The base is now √4, and the height is 1 . " Hypotenuse "=√((√4 )^2+1^2 )=√(4+1)=√5 The Pattern Because every new triangle uses the previous square root as its base and 1 as its height, squaring the base just removes the square root symbol, and we simply add 1 to the number inside. Therefore, the lengths of the hypotenuses in the spiral form a perfect, continuous sequence of square roots: √𝟐,√𝟑,√𝟒,√𝟓,√𝟔,√𝟕,√𝟖,√𝟗,√𝟏𝟎,√𝟏𝟏,… and so on! The 𝑛th triangle in the spiral will always have a hypotenuse of √(𝑛+1)