Find the lengths of the hypotenuses of all the right triangles in the given figure which is referred to as the square root spiral. Class 9
Find the lengths of the hypotenuses of all the right triangles in the given figure which is referred to as the square root spiral. Class 9
Question 1.
Find the lengths of the hypotenuses of all the right triangles in the given figure which is referred to as the square root spiral. Class 9
Solution:
In the square root spiral, each successive right triangle is formed such that:
One leg is always 1 unit
The other leg is the hypotenuse of the previous triangle.
Using Baudhayana - Pythagoras theorem, we find each new hypotenuse.
Triangle 1:
Legs are 1 unit and 1 unit
H1 = [latex]\sqrt{\left(1^2+1^2\right)}[/latex]
= [latex]\sqrt{1+1}[/latex] = [latex]\sqrt{2}[/latex] units.
Triangle 2:
Legs are [latex]\sqrt{2}[/latex] units and 1 unit
H2 = $\sqrt{\left((\sqrt{2})^2+1^2\right)}$
= $\sqrt{2+1}$ = [latex]\sqrt{3}[/latex] units
Triangle 3:
Legs are [latex]\sqrt{3}[/latex] units and 1 unit
H3 = $\sqrt{\left((\sqrt{3})^2+1^2\right)}$
= $\sqrt{3+1}$
= $\sqrt{4}$ = 2 units.
Triangle 4:
Legs are 2 units and 1 unit
H4 = $\sqrt{\left(2^2+1^2\right)}$
= $\sqrt{4+1}$ = $\sqrt{5}$ units.
Triangle 5:
Legs are $\sqrt{5}$ units and 1 unit
H5 = $\sqrt{\left((\sqrt{5})^2+1^2\right)}$
= $\sqrt{5+1}$ = $\sqrt{6}$ units.
Triangle 6:
Legs are $\sqrt{6}$ units and 1 unit
H6 = $\sqrt{\left((\sqrt{6})^2+1^2\right)}$ = $\sqrt{6+1}$ = $\sqrt{7}$ units.
Triangle 7:
Legs are $\sqrt{7}$ units and 1 unit
H7 = $=\sqrt{\left((\sqrt{7})^2+1^2\right)}$
= $\sqrt{7+1}$
= $\sqrt{8}$ = 2$\sqrt{2}$ units.
Triangle 8:
Legs are 2$\sqrt{2}$ units and 1 unit
H8 = $\sqrt{\left((2 \sqrt{2})^2+1^2\right)}$
= $\sqrt{8+1}$
= $\sqrt{9}$ = 3 unts.
Triangle 9:
Legs are 3 units and 1 unit
H9 = $\sqrt{\left(3^2+1^2\right)}$
= $\sqrt{9+1}$ = $\sqrt{10}$ units.
Triangle 10:
Legs are $\sqrt{10}$ units and 1 unit.
H10 = $\sqrt{\left((\sqrt{10})^2+1^2\right)}$
= $\sqrt{10+1}$ = $\sqrt{11}$ units.
The lengths of the hypotenuses are: $\sqrt{2}$ units, $\sqrt{3}$ units, 2 units, $\sqrt{5}$ units, $\sqrt{6}$ units, $\sqrt{7}$ units, 2$\sqrt{2}$ units, 3 units, $\sqrt{10}$ units, $\sqrt{11}$ units.