Question

Tell whether the given side lengths can represent the sides of a right triangle

10, 12, 20
Determine if the segment lengths form an acute, right, or obtuse triangle,
10, 11, 14
26.


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Answer 1

23. By the diagram, the triangle is a
`45^\circ - 45^\circ-90^\circ` triangle. Therefore, the triangle is isosceles, so
`x=6.` And by the Pythagorean theorem,
`y^2=x^2+6^2=6^2+6^2=72,` and taking the square root yields
`y = 6√(2).`

24. Notice that by the diagram, the triangle is a
`30^\circ-60^\circ-90^\circ` triangle. It is well known that in these special types of triangles, if
`a` is the side length opposite the
`30^\circ` angle, then the side length opposite the
`90^\circ` angle is
`2a,` and finally the side length opposite the
`60^\circ` is
`a√(3).` This is shown in the image I've attached below, and can also be easily proven by the fact that two of these triangles combine to form an equilateral triangle.

So using this fact, we know that
`x = (8)/(2) = 4,` and
`y = x√(3) = 4√(3).`

25. Tell whether the given side lengths can represent the sides of a right triangle: 10, 12, 20

For this question, we simply use the converse of the Pythagorean theorem. One part of its statement says that if a triangle has sides of length
`a, b, c` such that
`a^2 + b^2 = c^2`, and
`c` is the longest side, then the angle opposite the side of length
`c` is a right angle.

Thus, if the triple
`(a, b, c)` does not satisfy the equation, the triangle it forms is not a right triangle. In our case, the corresponding values for
`(a, b, c)` are
`(10, 12, 20),` because
`20` is the largest number in this triple. But
`10^2+12^2=244\\eq 400 = 20^2,` so the triangle with side lengths
`10, 12, 20` is not a right triangle.

26. Another statement of the converse of the Pythagorean theorem is that if a triangle has side lengths
`a, b, c` such that
`c^2 > a^2+b^2` , with
`c` being the longest side, then the triangle is an obtuse triangle. So the corresponding values for
`a, b, c` are
`10, 11, 14,` and because
`14^2 = 196 < 221 = 10^2+11^2,` the triangle is an obtuse triangle.