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# [Smart Math] Geometry Problem 3

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

### 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).

1. $\frac{-14}{5},-1$
2. $-1,\frac{-14}{5}$
3. $1,\frac{-14}{5}$
4. $\frac{-1}{5},-14$
5. $1,\frac{14}{5}$

### The Usual Method

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

$x=\frac{2(2)+3(-3)}{2+3}=\frac{4-9}{5}=-1$

$y=\frac{2(-1)+3(-4)}{2+3}=\frac{-2-12}{5}=\frac{-14}{5}$

Hence, coordinates of B are

$-1,\frac{-14}{5}$

.

(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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