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Chapter 1: Problem 26

Show that the points form the vertices of the indicated polygon. $$\text { Right triangle: }(-1,3),(3,5),(5,1)$$

Short Answer

Expert verified
To be filled after performing steps 1 to 3. If any of the Pythagorean theorem condition from step 2 is satisfied, then the points (-1,3), (3,5), and (5,1) form the vertices of a right triangle, otherwise not.

Step by step solution

01

Calculate the distance between points

The distance 'd' between two points (x鈧�, y鈧�) and (x鈧�, y鈧�) is given by \[d = \sqrt{(x鈧�-x鈧�)虏+(y鈧�-y鈧�)虏}\]. Therefore, the distance between the points (-1,3) and (3,5), let's call this 'a', is given by \[a = \sqrt{(3-(-1))虏 + (5-3)虏} = \sqrt{16+4} = 2\sqrt{5}\]. Similarly calculate the distance 'b' between the points (3,5) and (5,1), and the distance 'c' between the points (-1,3) and (5,1).
02

Check the Pythagorean theorem condition

According to the Pythagorean theorem, if these points form a right triangle then \(a虏 = b虏 + c虏\) or \(b虏 = c虏 + a虏\) or \(c虏 = a虏 + b虏\). Calculate \(a虏\), \(b虏\), and \(c虏\) and check if any of these conditions are satisfied.
03

Confirm the polygon

If any of the conditions from step 2 is satisfied, then the points form the vertices of a right triangle, otherwise not.

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