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

**Problem**

**Problem**

The coordinates of P, Q and R are , (1, –3) and (*x*, *y*) respectively. If R is the midpoint of PQ, find the values of *x *and *y*.

**The Usual Method**

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Using the midpoint formula:

Similarly,

Hence, answer is

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**(Ans: 3)**

*Estimated Time to arrive at the answer = 45 seconds.*

**Using Technique**

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Since the y coordinate of point Q is negative with a value greater than that of point P (i.e. 3 > , the sum of these ( ) will be negative. Thus the *y* coordinate of the answer should be negative and the *x* coordinate positive. This eliminates options ‘1’, ‘2’ and ‘4’.

Now looking at the coordinates of P and Q, we observe that the ‘*x*’ and ‘*y*’ coordinates of P are fractions with denominators 3 and 2 respectively, whereas the coordinates of Q are integers.

Hence, half of the sum of ‘x’ coordinates of P and Q would yield a fraction with denominator 3 x 2 = 6.

Similarly, half of the sum of y coordinates of P and Q would yield a fraction with denominator 2 x 2 = 4.

The only option that satisfies this condition is option ‘3’.

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**(Ans: 3)**

*Estimated Time to arrive at the answer = 10 seconds.*

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