Exercise 41 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 start by calculating the lengths of the segments formed by the given points. We will use 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 QR, RS, and QS. Then we can use a calculator to approximate the obtained square roots, if needed.

Points	Substitute	Simplify	Segment Length
Q(2,6) and R(6,5)	QR=sqrt((6-2)^2+(5-6)^2)	QR=sqrt(16+1)	QR≈ 4.1
R(6,5) and S(1,2)	RS=sqrt((1-6)^2+(2-5)^2)	RS=sqrt(25+9)	RS≈ 5.8
Q(2,6) and S(1,2)	QS=sqrt((1-2)^2+(2-6)^2)	QS=sqrt(1+16)	QS≈ 4.1

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 these side lengths satisfy the assumptions of the theorem. If yes, then it is possible to form a triangle with the given side lengths.

Sides	Triangle Inequality Theorem	Simplified Inequality	Is The Theorem Satisfied?
QR and RS	4.1+5.8? >4.1	9.9>4.1	Yes
RS and QS	5.8+4.1? >4.1	9.9>4.1	Yes
QR and QS	4.1+4.1? >5.8	8.2>5.8	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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