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# [Smart Math] Geometry Problem 1

Here’s and example of a SMART MATH problem for GEOMETRY. ### Problem

Find the distance between points P (-3, 2) and Q (-5, 8).

1. $\sqrt{40}$units
2. 4 units
3. 6 units
4. $\sqrt{60}$units
5. $\sqrt{30}$units

### The Usual Method

Using formula of distance between two points = $\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$

= $\sqrt{((-3)-(-5))^{2}+((2)-(8))^{2}}$

= $\sqrt{4+36}$

= $\sqrt{40}$ units

(Ans: 1)

Estimated Time to arrive at the answer = 30 seconds.

### Using Technique

Difference between $x_{1}$and $x_{2}$= 2 units
Difference between $y_{1}$and $y_{2}$= 6 units
Since, the lines PQ, $(x_{1}-x_{2})$ and $(y_{1}-y_{2})$form a right angled triangle with PQ as the hypotenuse. We already know that the hypotenuse of any right triangle should be more than the longest right side but less than the sum of the right sides. Thus length of PQ should be lying between 6 < PQ < (6 + 2). The only value satisfying this range is $\sqrt{40}$ $\approx$6.3 units.