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

**Problem**

**Problem**

Find the coordinates of the point B if B divides A and C internally in the ratio of 2 : 3. Given that A (–3, –4) and C (2, –1).

**The Usual Method**

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Let the coordinates of B be (*x*, *y*). Using the internal division formula, we get:

Hence, coordinates of B are

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**(Ans: 2)**

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

**Using Technique**

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If a line is cut by a point in a certain ratio, then its length along the X-axis and that along the Y-axis is also cut in the same ratio.

In this case, the *x* coordinate of A (–3) and that of C (2) are –3 –2 = –5 or 5 units apart along the X-axis. When this has to be divided in the ratio of 2 : 3; 2 parts towards point A and 3 parts towards point C, we get the coordinate of point B.

As can be seen in the above figure; the value of *x* coordinate of point B will be –1 (2 units on the right of –3 or 3 units on the left of 2). Once we know that the *x* coordinate is –1, we can look at the options and see that only option ‘2’ satisfies the requirement and hence the answer should be option ‘2’.

**(Ans: 2)**

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

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