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Can the Sides 9, 12, and 11 Form a Valid Triangle?

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The triangle inequality theorem states that for any three sides to form a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's explore if the sides 9, 12, and 11 can form such a triangle.

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Checking the Triangle Inequality

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To check if the sides 9, 12, and 11 can form a triangle, we will apply the triangle inequality theorem:

" "" "9 12 > 11" "9 11 > 12" "12 11 > 9" "" "

These conditions are met as follows:

" "" "21 > 11 (True)" "20 > 12 (True)" "23 > 9 (True)" "" "

Since all these conditions are satisfied, the sides 9, 12, and 11 can indeed form a valid triangle.

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Constructing a Triangle

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The triangle formed by sides 9, 12, and 11 can be constructed:

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Let's represent the triangle with sides a 9, b 12, and c 11. We can use the Law of Cosines to determine the angles of the triangle:

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c^2 a^2 b^2 - 2abcos(theta)

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For side c (12), we have:

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$12^2  9^2   11^2 - 2 cdot 9 cdot 11 cdot cos(theta)

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Simplifying:

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144 81 121 - 198cos(theta)

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144 202 - 198cos(theta)

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198cos(theta) 202 - 144

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cos(theta) 58 / 198

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cos(theta) 0.2939

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theta cos^{-1}(0.2939)

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theta approx 73.02^{circ}

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The triangle is valid with the angle opposite the side of length 12 being approximately 73.02°.

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Visualizing the Triangle

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As a fun exercise, let's determine the height of the triangle when it is standing on the side with length 8 units. Here are the steps:

" "" "Represent the triangle with the red side of length 10 units and the blue side of length 11 units." "Use the Pythagorean theorem to find the height:" "Let the height be h and the base be x." "The equation for the red side (10 units) is:" "

$h^2  10^2 - x^2$

" "The equation for the blue side (11 units) is:" "

$h^2  11^2 - (8 - x)^2$

" "Set the two equations equal to each other:" "

$100 - x^2  121 - (8 - x)^2$

" "Solve for x:" "$100 - x^2 121 - (64 - 16x x^2)$" "$100 - x^2 121 - 64 16x - x^2$" "$100 57 16x$" "$43 16x$" "$x 43/16 approx 2.69$" "$8 - x approx 5.31$" "The height h is then:" "

$h^2  100 - (43/16)^2 approx 100 - 23751/256 approx 23751/256$

" "$h approx sqrt{23751/256} approx 9.75$ units" "" "

This height is approximate and useful for visualizing the triangle.

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Conclusion

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Yes, the sides 9, 12, and 11 can form a valid triangle. They satisfy the triangle inequality, and we can construct the triangle with its angles determined using the Law of Cosines. The height of the triangle, when standing on the side with length 8 units, can be calculated using the Pythagorean theorem.

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