Represent \( \sqrt{6}, \sqrt{7}, \sqrt{8} \) on the Number Line
Construction Method:
To Construct \( \sqrt{6} \):
- Draw a number line and mark O (0) and A such that OA = 5 units.
- At point A, draw a perpendicular AB of length 1 unit.
- Join OB.
\[ OB = \sqrt{5^2 + 1^2} = \sqrt{6} \]
With O as center and radius OB, cut the number line at point P. Then OP = \( \sqrt{6} \).
To Construct \( \sqrt{7} \):
- Take OA = 6 units.
- Draw AB ⟂ OA of length 1 unit.
- Join OB.
\[ OB = \sqrt{6^2 + 1^2} = \sqrt{7} \]
Mark this length on the number line.
To Construct \( \sqrt{8} \):
- Take OA = 7 units.
- Draw AB ⟂ OA of length 1 unit.
- Join OB.
\[ OB = \sqrt{7^2 + 1^2} = \sqrt{8} \]
Mark this point on the number line.
Approximate Positions:
\[ \sqrt{6} \approx 2.45,\quad \sqrt{7} \approx 2.64,\quad \sqrt{8} \approx 2.83 \]
These points lie between 2 and 3 on the number line.
Final Result:
The numbers \( \sqrt{6}, \sqrt{7}, \sqrt{8} \) are represented on the number line using right triangle construction.