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How to solve a triangle formed by three (x,y) points.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find the unknown sides and angles of a triangle defined by three (x,y) points. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

Plot three points on the (x,y) plane and join them with lines. If the points are not in the same straight line, you will have created a triangle. This diagram shows the triangle created by the three points (1,3), (-2,-2) and (3,-1):Â

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This triangle has internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':

The distance formula can be used to find the distance between two (x,y) points.

For example, consider the points (x1, y1) and (x2, y2). The distance 'd' between them is given by the formula:

We can use the distance formula to find the distance between each pair of points making up our triangle. These distances are the lengths of the three sides.

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For the example above, the points are: (1,3), (-2,-2) and (3,-1), so we calculate the side lengths by taking these points two at a time:

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Side 1 joining (1,3) and (-2,-2)

Similarly, the side joiningÂ  (1,3) and (3,-1) is 4.47214, and

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  the side joining (-2,-2) and (3,-1)is 5.09902

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We now know the three side lengths of the triangle, as shown below:

Once the three side measurements are known, then the internal angles 'A', 'B' and 'C' can be found as well.

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When no angles are known, the cosine rule is the only option.Â

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Step 1: Begin by using the cosine rule to find the largest angle.Â

NOTE: We find the largest angle first, because there can only be one angle in a triangle that is obtuse (greater than 90°). If a triangle has an obtuse angle, then this will be it. The reason for finding it first is that in the next step we will use the sine rule to find the second angle. The inverse sin operation that we will use can only give us acute angles (less than 90°), so we avoid a possible wrong answer by first eliminating the only possibility of an obtuse angle.

The largest angle is always opposite to the largest side, so this is angle 'B' in this example ...

The cosine rule...

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Find the inverse cos of 0.263117 using a scientific calculator...

Â Â Â Â Â Â Â  B = cos-1(0.263117)

Â Â Â Â Â Â Â Â Â Â Â  = 74.7449°

Step 2: Use the sine rule to find one of the remaining angles.Â

NOTE: The sine rule is easier to use than the cosine rule.

To find angle 'C' with the sine rule:

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Find the inverse sin of 0.843661 using a scientific calculator...

Â Â Â Â Â Â Â  A = sin-1(0.843661)

Â Â Â Â Â Â Â Â Â Â Â  = 57.5288°

Step 3: Use the 'sum of internal angles' rule to find the third angle...Â

The sum of the internal angles equals 180º ...

Â Â Â Â Â Â Â  A + B + C = 180º

so

Â Â Â Â Â Â Â  B = 180º - (A+C)Â

Â Â Â Â Â Â Â Â Â Â Â  = 180° - (74.7449° + 57.5288°)

Â Â Â Â Â Â Â Â Â Â Â  = 180° - 132.274°

Â Â Â Â Â Â Â Â Â Â Â  = 47.7263°

The triangle is now solved. This diagram shows all of the sides and angles:

The Method section below shows you how Maths Helper Plus can easily solve your triangles, creating both a labelled diagram and full working steps.

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### Method:

Maths Helper Plus can solve a triangle given three (x,y) points. Full working steps and a labelled diagram are created. The steps below will show you how...

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NOTE: This document has already been set up to solve the example triangle as described in the 'theory' section of this topic.

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#### Step 2Â  Display the triangle solver options box

Double click the mouse in the border to the left of the calculations. ( This area is shaded pale blue in the diagram below.) The triangle solver options box will display its '3 Points' tab...

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Click the 'Clear' button to remove the previous triangle, then click on the edit box near angle 'A' (lower left). Type the (x,y) pointÂ  for the lower left point of your triangle. Repeat for the other two points.

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HINT: Try to ensure that the points entered as lower left, lower right and top really are in these approximate positions on the graph, otherwise the labels drawn on the diagram will be poorly positioned.

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Click the 'OK' button to close the options box. The calculations and triangle diagram will be displayed on your screen.

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#### Step 3Â  Adjust the size of the diagram

If the triangle diagram is too big to display properly on your computer screen, briefly press the F10 key to reduce its size. To make the diagram bigger, hold down a Ctrl key while you press F10.

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