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In quadrilateral $ABCD$, we have $AB=3,$ $BC=6,$ $CD=4,$ and $DA=4$. If the length of diagonal $AC$ is an integer, what are all the possible values for $AC$? Explain your answer in complete sentences. Hint(s): Draw a picture! Apply the triangle inequality. There is more than one triangle to think about.

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sqdancefan

Answer:

AC ∈ {4, 5, 6, 7}

Explanation:

When two sides of a triangle are specified, the allowable values for the length of the third side are between the sum and difference of the given sides.

In ∆ACD, sides DA and CD are both given as length 4. Thus the possible range of values for side AC is 0 – 8. The extremes of this range result in ∆ACD having zero area, so we assume they are not of interest.

In ∆ACB, sides AB and BC are given as having lengths 3 and 6. Thus the possible range of values for side AC is 3 – 9. The extremes of this range result in ∆ACB having zero area, so we assume they are not of interest.

The integers that are in both ranges and that give triangles with non-zero area are ...

... AC ∈ {4, 5, 6, 7}

Answer:

4, 5, 6, 7

Explanation:    

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