Question 1177021
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Congruent diagonals means the trapezoid is isosceles, and triangles ABX and CDX are both isosceles.<br>
The bases of triangles ABX and CDX are 30 and 54, so the ratio of similarity between the two triangles is 30:54 = 5:9.<br>
That means the ratio of corresponding sides AX and CX is 5:9.<br>
And since the length of AC is 56, that makes AX=20 and CX=36.<br>
Triangle ABC is then isosceles with legs 20 and base 30.<br>
A perpendicular from X to side AB divides triangle ABC into two congruent right triangles with hypotenuse 20 and one leg 30/2=15.<br>
Use the Pythagorean Theorem to find the height of triangle ABC; then use that height and the base to find the area of the triangle using area = one-half base times height.<br>