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McGraw Hill Integrated II, 2012
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McGraw Hill Integrated II, 2012 View details
5. The Triangle Inequality
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Exercise 39 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 FG, GH, and FH. Then we can use a calculator to approximate the obtained square roots, if needed.
Points Substitute Simplify Segment Length
F(-4,3) and G(3,-3) FG=sqrt((3-(-4))^2+(-3-3)^2) FG=sqrt(49+36) FG≈ 9.2
G(3,-3) and H(4,6) GH=sqrt((4-3)^2+(6-(-3))^2) GH=sqrt(1+81) GH≈ 9.1
F(-4,3) and H(4,6) FH=sqrt((4-(-4))^2+(6-3)^2) FH=sqrt(64+9) FH≈ 8.5

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?
FG and GH 9.2+9.1? >8.5 18.3>8.5 Yes
GH and FH 9.1+8.5? >9.2 17.6>9.2 Yes
FG and FH 9.2+8.5? >9.1 17.7>9.1 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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