In a trapezoid where AD||BC and AC perpendicular to CD, and AC bisects angle BAD, the relationship between the lengths specified is that the lengths of AC and BD are equal.
Given that trapezoid ABCD has AD||BC, AC perpendicular to CD, AC bisects angle BAD, and the area of ABCD is 42, we can find the relationship between the lengths and angles specified in the trapezoid.
Since AC bisects angle BAD, it divides the trapezoid into two congruent triangles: Triangle BAC and Triangle DAC.
This means that the areas of Triangle BAC and Triangle DAC are equal.
Let h be the height of the trapezoid and x be the length of CD.
The area of a trapezoid is given by A = 1/2 * (sum of bases) * height.
In this case, the sum of bases is AC + BD, and the height is h. So we have:
42 = (1/2) * (AC + BD) * h
Since AC is the same length as BD in a trapezoid where its legs are parallel, and since AC = BD, we can simplify the equation to:
42 = (1/2) * 2 * AC * h
42 = AC * h
Now, since AC bisects angle BAD, we can conclude that Triangle BAC and Triangle DAC are congruent.
So the lengths of AC and BD are equal.
Therefore, we can substitute AC = BD into the equation:
42 = AC * h = BD * h
So the relationship between the lengths specified in the trapezoid is that the lengths of AC and BD are equal.