Ayush travels House \(\to\) Bank \(\to\) School \(\to\) Office along straight segments instead of going directly House \(\to\) Office. If \(H(2,4),\ B(5,8),\ S(13,14),\ O(13,26)\), find the extra distance travelled.
Extra distance \(= 27 - \sqrt{605}\;\text{km} \approx 2.404\;\text{km}.\)
Step 1: Recall distance formula.
If two points are \((x_1,y_1)\) and \((x_2,y_2)\), then
\[ \,\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
Step 2: Calculate distance from House to Bank (H to B).
H(2,4), B(5,8)
\[ HB = \sqrt{(5-2)^2 + (8-4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\;\text{km} \]
Step 3: Calculate distance from Bank to School (B to S).
B(5,8), S(13,14)
\[ BS = \sqrt{(13-5)^2 + (14-8)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10\;\text{km} \]
Step 4: Calculate distance from School to Office (S to O).
S(13,14), O(13,26)
Here, the x-coordinates are the same, so the line is vertical.
\[ SO = |26 - 14| = 12\;\text{km} \]
Step 5: Add up the three parts.
\[ \text{Total distance via Bank and School} = HB + BS + SO = 5 + 10 + 12 = 27\;\text{km} \]
Step 6: Calculate direct distance from House to Office (H to O).
H(2,4), O(13,26)
\[ HO = \sqrt{(13-2)^2 + (26-4)^2} = \sqrt{11^2 + 22^2} = \sqrt{121 + 484} = \sqrt{605}\;\text{km} \]
Using calculator: \(\sqrt{605} \approx 24.596\;\text{km}\).
Step 7: Find the extra distance travelled.
\[ \text{Extra distance} = (27 - \sqrt{605})\;\text{km} \approx (27 - 24.596)\;\text{km} = 2.404\;\text{km} \]
Final Answer: Ayush travels about 2.404 km more when he goes through Bank and School instead of going directly from House to Office.