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

### Geometry Problem 1

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

- $latex \sqrt{40}$units
- 4 units
- 6 units
- $latex \sqrt{60}$units
- $latex \sqrt{30}$units

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### The Usual Way

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

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

= $latex \sqrt{4+36}$

= $latex \sqrt{40}$ units

**(Ans: 1)**

*Estimated Time to arrive at the answer = 30 seconds.*

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### The Smart Way

Difference between $latex x_{1}$and $latex x_{2}$= 2 units

Difference between $latex y_{1}$and $latex y_{2}$= 6 units

Since, the lines PQ, $latex (x_{1}-x_{2})$ and $latex (y_{1}-y_{2})$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 $latex \sqrt{40}$ $latex \approx $6.3 units.

**(Ans: 1)**

*Estimated Time to arrive at the answer = 15 seconds.*

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