Programmed tutorial: Type of a 3D Triangle: Right Triangle

P(3, 1, 4);  Q(1, 5, -1);  R(3, 5, -1)

They define a triangle. 

We want to determine if it is a right triangle, an equilateral triangle, an isosceles triangle, or a more general triangle.

First, let’s plot the points on the usual axes (the x-axis slanting out of the plane of the diagram).  Draw those axes on paper, and then click “Next” to check your diagram.

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Here are the axes, with a general point (not P, Q, or R) shown.

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Notice that we measure x in blue in the x-direction (out of the yz-plane).

Then we measure y in yellow in the y-direction, and finally a distance z in green in the z-direction.

We’ll do our diagram in steps.  Start your diagram on paper with the point P(3, 1, 4) and click “Next” to check.

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Add Q(1, 5, -1) to your diagram and check by clicking “Next”.

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Note the negative value of Q_z plotted downward in green.

Add R(3, 5, -1) to your diagram and check by clicking “Next”.

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Add the triangle to your diagram.  Then click “Next” to check.

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How can we check what type of triangle we have here?

We need the lengths of the sides of the triangle.  Then we can see if three or two sides are equal (for equilateral or isosceles), or whether a² + b² = c² as in a right triangle.  If none of these are true, then we have a general triangle.

How can we find the lengths of the sides?

We can apply the 3D form of the distance equation to each pair of points.

Set that up for the distance First use the symbols .

We get
This result involves a square root.  Since we are comparing distances for equality, we can also use the square of the distance and avoid the square root.  Furthermore, for use in the Pythagorean Theorem, we need the squares of the lengths anyway.  So from here on, let’s use the squares, such as

Now substitute the data for Px, Py, etc.

Notice how the negative value of Q_z is handled.

Simplify this equation.

Set up with its data.

Simplify this equation.

Set up with its data.

Simplify this equation.

To summarize, we have

Is this an equilateral or isosceles triangle?

No, all three sides are different in length.

To test if it is a right triangle, how do we decide which side is a, b, or c in a² + b² = c²?

We just use the longest side (biggest square) to get c².

Is this a right triangle?

Yes:
, or
45 = 4 + 41.

Please see other cases by using the Detailed Contents link below.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor