In a rectangle, the bisector of the right angle divides the side into 42 cm and 14 cm

In a rectangle, the bisector of the right angle divides the side into 42 cm and 14 cm segments. In which segments does this bisector bisect the diagonal?

Let us denote the rectangle by the letters ABCD, SK is the bisector of angle C, DK = 42 cm, AK = 14 cm. Let E be the point of intersection of the bisector of SK and diagonal BD.

Side AD is equal to the sum of the segments AK and DK, AD = 42 + 14 = 56 cm. BC = AD = 56 cm (opposite sides in the rectangle are equal).

The triangle ВСD is rectangular, we calculate the length of the diagonal according to the Pythagorean theorem: ВD = √ (56 ^ 2 + 42 ^ 2) = √ (3136 + 1764) = √4900 = 70 cm.

Consider the triangles BEC and DЕК: the angle BEC is equal to the angle DЕК (vertical angles), the angle BSE is equal to the angle DКЕ (internal cross-lying angles with parallel BC and AD and secant CK). Hence, the triangles are similar.

Let’s calculate the coefficient of similarity: k = ВС / DК = 56/42 = 4/3.

Hence, BE refers to DE as 4/3.

Let BE = 4x and DE = 3x. The length BD is 70 cm, we make the equation:

4x + 3x = 70;

7x = 70;

x = 10.

Hence, BE = 4 * 10 = 40 cm. DE = 3 * 10 = 30 cm.

Answer: the bisector divides the diagonal into 40 cm and 30 cm segments.