In a rectangular trapezoid, the large diagonal with a length of 24 is the bisector of an acute angle.
In a rectangular trapezoid, the large diagonal with a length of 24 is the bisector of an acute angle. find the area of a trapezoid if the distance from the apex of the obtuse angle to the diagonal is 9.
Let there be a trapezoid ABCD and DE – the height of the large diagonal (picture below)
We denote the angle (CAB) = a
Because AC – bisector, then angle (CAD) = a
Because the sum of the angles of the triangle AED = 180 and the angle (AED) = 90, then the angle (ADE) = 180 – 90 – a = 90 – a
Because the sum of the angles of the triangle ACB = 180 and the angle (ABC) = 90, then the angle (ACB) = 180 – 90 – a = 90 – a
angle (ECD) = angle (BCD) – angle (ACB) = a
Because the sum of the angles of the triangle EDC = 180 and the angle (DEC) = 90, then the angle (EDC) = 180 – 90 – a = 90 – a
Triangles ADE and CDE are similar because their angles are equal
Triangles ADE and CDE are equal because they are similar and the side DE is common, opposite the angle a
Then AE = EC = 24/2 = 12 cm
The area of right-angled triangles AED and DEC is 9 * 12/2 = 54
Triangles ABC and CED are similar because their angles are equal
BA / CB = CE / ED = 12/9 = 4/3
Let CB = 3x, AB = 4x
By the Pythagorean theorem AB ^ + CB ^ = AC ^
9x ^ + 16x ^ = 24 ^
x ^ = 24 ^ / 25
x = 24/5
The area of triangle ABC is AB * BC / 2 = 3 * 4 * 24 / (5 * 2) = 144/5
The area of the trapezoid is equal to the sum of the areas of the triangles.
S = 54 + 54 + 28.8 = 136.8.
Answer: 136.8.
