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# [Smart Math] Geometry Problem 5

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

ABC is a triangle with vertices A (4, 8), B (7, 3) and C (3, 1). Find the equation of the median AD.

1. x + y = 7
2. x + y = 10
3. 3x + y = 10
4. 2x + y = 16
5. 6x + y = 32

### The Usual Method

The first step is to get the coordinates of point D of the median AD. Since, AD is the median; point D is the mid point of BC. $\therefore$ D = $\left( \frac{7+3}{2},\frac{3+1}{2} \right)$ = (5, 2)

Now, using the coordinates of point A and D in the Two-point form of equation of a straight line, we get: $\left( y-8 \right)=\left( \frac{8-2}{4-5} \right)\left( x-4 \right)$ $\therefore$ $\left( y-8 \right)$= –6 $\left( x-4 \right)$ $\therefore$6x + y = 32 is the equation of the median AD.

(Ans: 5)

Estimated Time to arrive at the answer = 60 seconds.

### Using Technique

Since, the median includes point A, the coordinates of point A should satisfy the equation of the median. Hence, just check which amongst the equations is satisfied by the coordinates of point A (4, 8).

x + y = 4 + 8 = 12 $\ne$7

x + y = 4 + 8 = 12 $\ne$10

3x + y = 3 x 4 + 8 = 20 $\ne$10

2x + y = 2 x 4 + 8 = 16 =16

6x + y = 6 x 4 + 8 = 32 =32

As can be seen that only options‘d and ‘5’ are satisfied by the coordinates of point A, thus we can eliminate options ‘1’, ‘2’ and ‘3’.

Since, we are left with two options, we need to do further elimination. This can be done by substituting the coordinates of point D.

Coordinates of point D can be found mentally as it is the average value of the corresponding coordinates of point B and C (D being the midpoint of B and C).

Now, checking between the options’4’ and ‘5’ which one is getting satisfied with the coordinates of point D (5, 2).

2x + y = 2×5 + 2 = 12 $\ne$16

6x + y = 6×5 + 2 = 32 =32

Hence the answer is option ‘5’.

(Ans: 5)

Estimated Time to arrive at the answer = 30 seconds.

(Note: It would take about 15 seconds to get to the answer if we are able to eliminate all but one option from the list in the first round of elimination itself. But since, we need to do another round of elimination; an additional 15 seconds are added)

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