# In rectangle ABCD, side BC = 80. Through points M and K, which belong to sides AB and BC relatively

**In rectangle ABCD, side BC = 80. Through points M and K, which belong to sides AB and BC relatively, a straight line is drawn parallel to the diagonal AC = 100. Find the length of the larger side of a triangle MBK if BK = 20.**

Right-angled triangles BKM and ABC are similar in acute angle, since, by condition, KM is parallel to AC, and then the angle BCA = BKM as the corresponding angles at the intersection of parallel straight lines. Then the coefficient of similarity is: K = BC / BC = 80/20 = 4.

AC / MK = 4.

MK = AC / 4 = 100/4 = 25 cm.

In a right-angled triangle BKC, according to the Pythagorean theorem, BM ^ 2 = MK ^ 2 – BK ^ 2 = 625 – 400 = 225.

BM = 15 cm.

Let the length of the segment AM = X cm, then AB = (X + BM) = (X + 15) cm.

Then in a right-angled triangle ABC, according to the Pythagorean theorem, AB ^ 2 = AC ^ 2 – BC ^ 2.

(X + 15) ^ 2 = 10000 – 6400.

X ^ 2 + 30 * X + 225 = 3600.

X ^ 2 + 30 * X – 3375 = 0.

Let’s solve the quadratic equation.

X = 45 cm.

Then AB = 45 + 15 = 60 cm.

Answer: The length of the longer side is 80 cm, the length of the smaller side is 60 cm.