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.

AC ∈ {4, 5, 6, 7}

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}

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