Question

Match each set of vertices with the type of triangle they form.

A(2, 0), B(3, 2), C(5, 1)
obtuse scalene triangle
A(4, 2), B(6, 2), C(5, 3.73)
isosceles right triangle
A(-5, 2), B(-4, 4), C(-2, 2)
right triangle
A(-3, 1), B(-3, 4), C(-1, 1)
acute scalene triangle
A(-4, 2), B(-2, 4), C(-1, 4)

Answer 1

9514 1404 393

Answer:

• right
• acute
• obtuse
• right
• obtuse

Explanation:

When the same problem is repeated, I like to solve it using a spreadsheet. That way, the formulas only need to be entered once, and the arithmetic is (almost) guaranteed to be done correctly.

A "form factor" computed from side lengths can be used to determine the type of triangle. Where 'c' is the long side, that factor can be computed as ...

f = a² +b² -c²

and interpreted as follows:

• f = 0, right triangle
• f > 0, acute triangle
• f < 0, obtuse triangle

(The sign of f matches the sign of the cosine of the largest angle computed using the law of cosines.)

Of course, a right triangle can also be identified by looking at the slopes of the sides of the triangle. If any pair of slopes has a product that is -1, or if any pair is 0 and "undefined", then the triangle will be a right triangle.

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The attached spreadsheet is designed to accommodate a number of different problem requirements. It shows both side lengths and slopes, and it shows the "form factor" as described above. The final classification is shown at far right.