Question

PLEASEEEE HELPPP!!!!!!WILL GIVE BRAINLIEST!!!!!!!!!!!! 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.

Answer 1

Answer:

AC = { 4, 5, 6, 7 }

Step-by-step explanation:

If you see, the diagonal AC forms two triangles, Δ ABC, and Δ ADC. In Δ ABC, AC = 3 units and BC = 6 units, while AC is yet to be known. Respectively in Δ ADC, AD = 4 units and CD = 4 units, while AC is again yet to be known.

In both triangles the triangle inequality can help find the possible value( s ) of AD, as this inequality only restricts some of the possible values with which AC can take. At the same time AC is shared among the two triangles, so if we can apply the Triangle Inequality to both of these triangles, the value of AC can be " further restricted. "

$Triangle 1. BC - AB < AC < BC + AB,\\6 - 3 < AC < 6 + 3,\\3 < AC < 9\\\\Triangle2. AD - CD < AC < AD + CD,\\4 - 4 < AC < 4 + 4,\\0 < AC < 8$

And there we have two inequalities, 3 < AC < 9, and 0 < AC < 8. Combining both inequalities the only possible integer values for AC would be 4, 5, 6, and 7.