# Find the square of the distance

• MHB
• anemone
In summary, the formula for finding the square of the distance is (x2-x1)^2 + (y2-y1)^2, also known as the Pythagorean theorem. This is important in science for calculating distances between points in space. The square of the distance is different from the distance because it gives the area between two points rather than just the length. It cannot be a negative number as it is always squared. Real-life applications include calculating distances between cities, finding diagonal lengths, and determining distances between objects in space.
anemone
Gold Member
MHB
POTW Director
The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance from $B$ to the center of the circle.

[TIKZ]
\draw[purple,thick] (0,0) arc (70:328:3);
\coordinate[label=above:A] (A) at (0,0);
\coordinate[label=left:B] (B) at (0,-4.43);
\coordinate[label=right:C] (C) at (1.48,-4.43);
\draw[thick] (A) -- (B) -- (C);
[/TIKZ]

[TIKZ][scale=1.3]
\draw[purple,thick] (0,0) arc (70:328:3);
\coordinate[label=above:$A$] (A) at (0,0);
\coordinate[label=left:$B$] (B) at (0,-4.43);
\coordinate[label=right:$C$] (C) at (1.48,-4.43);
\coordinate[label=left:$O$] (O) at (-1.03,-2.82);
\coordinate[label=right:$N$] (N) at (0.74,-2.21);
\draw[thick] (A) -- (B) -- (C);
\draw (C) -- (A) -- (O) -- (B) ;
\draw (N) -- (O) ;
\draw (0.12,-0.8) node{$\beta$} ;
\draw (-0.15,-0.8) node{$\alpha$} ;
[/TIKZ]
In the diagram, $O$ is the centre of the circle, $N$ is the foot of the perpendicular from $O$ to $AC$, and the angles $\alpha$, $\beta$ are as shown.

By Pythagoras, $AC = \sqrt{40}$, so $AN = \sqrt{10}$. By Pythagoras again, $ON = \sqrt{40}$. Then $\tan\beta = \frac13$ and $\tan(\alpha + \beta) = 2$. Therefore $$\tan\alpha = \frac{2-\frac13}{1 + \frac23} = 1$$ and so $\alpha = 45^\circ$.

Now use the cosine rule in triangle $OAB$ to get $OB^2 = 50 + 36 - 2*\sqrt{50}*6*\frac1{\sqrt2} = 86 - 60 = 26.$

## 1. How do you find the square of the distance between two points?

To find the square of the distance between two points, you first need to calculate the distance between the points using the distance formula. Then, you can simply square the result to find the square of the distance.

## 2. Why is it important to find the square of the distance?

Finding the square of the distance is important in many mathematical and scientific calculations. It allows us to measure and compare distances accurately, and is also used in various equations and formulas in fields such as physics, engineering, and geometry.

## 3. Can you explain the difference between distance and square of distance?

Distance is the measurement of how far apart two points are, while the square of distance is the result of multiplying the distance by itself. In other words, the square of distance is the area of a square with sides equal to the distance between the two points.

## 4. Are there any real-life applications for finding the square of the distance?

Yes, there are many real-life applications for finding the square of the distance. For example, it is used in navigation systems to calculate the distance between two locations, in physics to determine the force between two objects, and in geometry to find the length of the diagonal of a square.

## 5. Is there a specific formula for finding the square of the distance?

Yes, the formula for finding the square of the distance is (x2 - x1)^2 + (y2 - y1)^2, where (x1, y1) and (x2, y2) are the coordinates of the two points. This is known as the distance formula and is derived from the Pythagorean theorem.

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