# How to find whether a point lies inside a triangle or not

You might have solved last year CAT papers(when it was available) or any AIMCAT/SIMCAT/..CAT.

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A type of question which is asked is “whether a point lies inside a triangle or not?”

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If the coordinates are simple, you can easily make a coordinate system and determine if the point lies inside or outside of a triangle.

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But what if the coordinates are not so easy or having a big scale and you are not able to draw it in coordinate system so easily, or it is taking too much of time to draw.

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There is a very simple method, though a bit lengthy to solve this problem.

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**For e.g., If P(6,7) , Q(2,3) and R(4,-2) be the vertices of a triangle, then which of the following is not a point contained in this triangle ?**

**(a.) (4,3) (b.) (3,3) (c.) (4,2) (d.) (6,1)**

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Since the coordinates are simple, one can easily draw the triangle on paper by using coordinates and determine which of the point is not contained in the triangle.

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But what if I say, don’t draw it and do it without drawing the triangle on coordinate system.

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Looks really tough. But don’t worry we will provide you step by step approach to solve this problem in a very easy manner.

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Step 1 :Find the area of traingle PQR .

**Area of PQR = 1/2 x |[6(3+2)+2(-2-7)+4(7-3)]| = 1/2 x [30 -18+16] = 14 sq unit**

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Step 2: Let the point in the option to be checked be A. Find area of triangle PQA , QRA, PRA

**Considering 1st point (4,3), **

**Area tr. PQA = 1/2 x |[6(3-3)+2(3-7)+4(7-3)| = 1/2 x |[-8 +16]| = 4 sq unit.**

**Area tr. QRA = 1/2 x |[2(-2-3)+4(3-3) +4(3+2)]| = 1/2 x |[-10+20]| = 5 sq unit.**

**Area tr. PRA = 1/2 x |[6(-2-3)+4(3-7)+4(7+2)]| = 1/2 x |[-30 -16 +36] = 5 sq unit.**

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Step 3: If the point is inside the triangle, sum of areas of the three triangles = Area of original triangle.

**Here, Area tr. PQA + Area tr. QRA + Area tr. PRA = 14 sq unit = Area Tr. PQR **

**So Point A (4,3) is inside the triangle PQR.**

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Step 4: If the point is outside the triangle, sum of areas of the three triangles =/= Area of original triangle.

According to step 2, we solve for all the 4 points and find that point (6,1) lies outside the triangle PQR.