Through the apex of the obtuse angle B of the parallelogram ABCD, the height BK is drawn to the side AD

| April 28, 2021 | education

Through the apex of the obtuse angle B of the parallelogram ABCD, the height BK is drawn to the side AD, AB = 9cm, AK = 6cm, DK = 2cm. Calculate the length of the projection of the side BC onto the straight line CD

Since BK is the height of the parallelogram, the angle of AKB is straight, and the triangle ABK is rectangular. Let us define the leg BK by the Pythagorean theorem. BK ^ 2 = AB ^ 2 – AK ^ 2 = 9 ^ 2 – 6 ^ 2 = 81 – 36 = 45.BK = 3 * √5.

Let us determine the area of ​​the parallelogram through the base AD and the height BK.

S = AD * BK = (AK + KD) * BK = (6 + 2) * 3 * √5 = 24 * √5 cm2.

The projection of the BC side to the CD side will be the EC segment obtained by drawing a perpendicular from the vertex B to the CD side. Since BK is also the height of the trapezoid to the side of CD, the area of ​​the trapezoid can be determined by the formula: S = CD * BE.

Then CD * BE = 24 * √5.

BE = 24 * √5 / 9 = 8 * √5 / 3.

Consider a right-angled triangle BEC, whose angle E is straight, then by the Pythagorean theorem CE ^ 2 = BC ^ 2 – BE ^ 2 = 82 – (8 * √5 / 3) 2 = 64 – 64 * 5/9 = 256/9 …

CE = 16/9 = 8/3 = 2 (2/3) cm.

Answer: The length of the projection of the BC on the CD is 2 (2/3) cm.



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