# Geometry Problem 2

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

### Geometry Problem 2

The coordinates of P, Q and R are $\left( \frac{2}{3},\frac{3}{2} \right)$, (1, –3) and (xy) respectively. If R is the midpoint of PQ, find the values of and y.

1. $\frac{-5}{6},\frac{-3}{4}$
2. $\frac{-3}{4},\frac{5}{3}$
3. $\frac{5}{6},\frac{-3}{4}$
4. $\frac{3}{4},\frac{5}{3}$
5. $\frac{3}{4},\frac{-5}{3}$

### The Usual Way

Using the midpoint formula:

$x=\frac{\frac{2}{3}+1}{2}=\frac{\frac{5}{3}}{2}=\frac{5}{6}$

Similarly,

$y=\frac{\frac{3}{2}+(-3)}{2}=\frac{\frac{-3}{2}}{2}=\frac{-3}{4}$

Hence, answer is $\frac{5}{6},\frac{-3}{4}$

(Ans: 3)

Estimated Time to arrive at the answer = 45 seconds.

### The Smart Way

Since the y coordinate of point Q is negative with a value greater than that of point P (i.e. 3 > $\frac{3}{2}$, the sum of these ( $-3+\frac{3}{2}$) 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’.

(Ans: 3)

Estimated Time to arrive at the answer = 10 seconds.

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