Exercise 38 Page 451

Use the Distance Formula to calculate the lengths of the segments formed by the given points.

Yes, see solution.

Practice makes perfect

Let's solve this exercise beginning by calculating the lengths of the segments formed by the given points. We will use something we are already familiar with, that is the Distance Formula. d=sqrt((x_2-x_1)^2+(y_2-y_1)^2) We can substitute the coordinates of the given points and calculate the lengths of XY, YZ, and XZ.

Points	Substitute	Simplify	Segment Length
X(1,- 3) and Y(6,1)	XY=sqrt((6-1)^2+(1-(- 3))^2)	XY=sqrt(25+16)	XY≈ 6.4
Y(6,1) and Z(2,2)	YZ=sqrt((2-6)^2+(2-1)^2)	YZ=sqrt(16+1)	YZ≈ 4.1
X(1,- 3) and Z(2,2)	XZ=sqrt((2-1)^2+(2-(- 3))^2)	XZ=sqrt(1+25)	XZ≈ 5.1

Now we will use the Triangle Inequality Theorem.

Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's verify if this theorem is true for these side lengths. If it is, then it is possible to form a triangle with the given side lengths.

Sides	Triangle Inequality Theorem	Simplified Inequality	Is the Theorem Satisfied?
XY and YZ	6.4+4.1? >5.1	10.5>5.1	Yes.
YZ and XZ	4.1+5.1? >6.4	9.2>6.4	Yes.
XY and XZ	6.4+5.1? >4.1	11.5>4.1	Yes.

As we can see, the theorem is satisfied for each pair of sides. Therefore, the given coordinates are the vertices of a triangle.

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