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# [Smart Math] Geometry Problem 4

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

Find the ratio in which the point X $\left( \frac{4}{3},4 \right)$ divides the line A (3, 5) and B (–2, 2).

1. 1 : 4
2. 1 : 1
3. 3 : 2
4. 1 : 2
5. 2 : 3

### The Usual Method

Assuming the ratio to be 1 : n and using the formula to find the coordinates of point of intersection; we get: $\frac{4}{3}=\frac{(-2)1+n(3)}{1+n}$

4 + 4n = -6 + 9n $\therefore$5n = 10 $\therefore$n = 2

Hence, ratio = 1 : 2.

You could also use the y coordinate as shown below: $4=\frac{1(2)+n(5)}{1+n}$

4 + 4n = 2 + 5n $\therefore$n = 2

Hence, ratio = 1 : 2.

(Ans: 4)

Estimated Time to arrive at the answer = 45 seconds.

### Using Technique

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 we choose the y coordinate since the coordinates are all integers and hence easy to work with.

Length of AX along Y-axis = 5 – 4 = 1

Length of XB along Y-axis = 4 – 2 = 2

Hence, the ratio = ratio of length of AX to that of XB = 1 : 2 i.e option ‘4’.

(Ans: 4)
Estimated Time to arrive at the answer = 10 seconds.

(Note: there is another shortcut here which is shorter than the one shown above.
If you look at the x coordinate of X; it is a fraction with denominator 3. Now, look at the options and find that option whose sum of Numerator and Denominator is 3. In this case, the option is ‘4’ which is the answer.)

Estimated Time to arrive at the answer = 5 seconds.

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