Shortest Distance Calculation for Cyclist Travel
The cyclist's journey forms a path that can be visualized geometrically.
- The cyclist starts at point $X$.
- Travels 3 km straight towards the east.
- Turns right (which means heading south).
- Travels 4 km towards the south.
- Reaches point $Y$.
This path creates a right-angled triangle, where:
- The eastward journey (3 km) is one leg of the triangle.
- The southward journey (4 km) is the other leg of the triangle.
- The shortest distance between point $X$ and point $Y$ is the hypotenuse of this triangle.
Applying Pythagorean Theorem
We use the Pythagorean theorem to find the length of the hypotenuse ($c$), given the lengths of the two legs ($a$ and $b$). The theorem states:
$a^2 + b^2 = c^2$
Here, $a = 3$ km and $b = 4$ km.
Step-by-Step Calculation
- Substitute the values of the legs into the formula:
$3^2 + 4^2 = c^2$
- Calculate the squares:
$9 + 16 = c^2$
- Add the values:
$25 = c^2$
- Find the square root to get the length of the hypotenuse:
$c = \sqrt{25}$
$c = 5$
Therefore, the shortest distance between point $X$ and $Y$ is 5 km.
Important Questions from Problem Solving
A tight fitting band is wrapped around the Equator. Another circular band whose length is 15 m more lies at a certain height over the first band. A group of human beings attempt to pass under the longer band. Can they walk under it?
(Earth’s circumference is roughly 40000 km. The height of human beings is between 1 & 2m)
Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(84, 73, 157) (96, 88, 184)
- Read the given paragraph and answer the question.
There are five persons A, G, M, J and D. Every person has to select one item each from a box containing items numbered from 1 to 5. A and J selected items 5 and 2 respectively. G cannot select item 3, while M cannot select item 3 or 5.
Which item can G select? - Read the given information carefully and answer the question that follows.
- There are four persons: P, Q, R and S.
- One of them is a doctor and plays football and cricket.
- P and Q are teachers.
- P plays badminton.
- The lawyer and both the teachers are swimmers.
- One of them also plays tennis.
- S plays carrom and is a lawyer.
- All the four persons play two games each and follow one profession.
The rectangle represents a set of calculations that are common across all the given equations. Identify the calculations involved and solve the fourth equation on the same basis:
$7 \blacksquare 4 = 27/4$
$6 \blacksquare 1 = 5$
$4 \blacksquare 3 = 11/3$
$a \blacksquare b = ?$
