The bisector of the right angle of a right-angled triangle divides the hypotenuse in a ratio of 4: 3.

| December 29, 2020 | education

The bisector of the right angle of a right-angled triangle divides the hypotenuse in a ratio of 4: 3. Calculate the length of this bisector if the perimeter of the triangle is 21√2.

Let the length of the segment AM = 3 * X cm, then, by condition, the length of the segment BM = 4 * X cm.
AM + BM = AB = 35 cm.
3 * X + 4 * X = 35.
7 * X = 35.
X = 35/7 = 5.
Then AM = 3 * 5 = 15 cm, BM = 4 * 5 = 20 cm.
By the property of the bisector of a triangle:
AC / AM = BC / VM.
AC / 15 = BC / 20.
AC / BC = 3/4.
Let the length of AC = 3 * Y cm, then BC = 4 * X cm.
By the Pythagorean theorem: AB2 = AC2 + BC2.
49 * X2 = 9 * Y2 + 16 * Y2 = 25 * Y2.
7 * X = 5 * Y.
Y = 7 * X / 5.
Then AC = 21 * X / 5, BC = 28 * X / 5.
Ravs = AC + BC + AB = 21 * √2 cm.
21 * X / 5 + 28 * X / 5 + 7 * X = 21 * √2.
84 * X = 105 * √2.
X = 105 * √2 / 84 = 5 * √2 / 4.
AC = 21 * (5 * √2 / 4) / 5 = 21 * √2 / 4 cm.
BC = 28 * (5 * √2 / 4) / 5 = 28 * √2 / 4 cm.
Then CM = √2 * (AC * BC / (AC + B)) = √2 * (21 * √2 / 4 + 28 * √2 / 4) / (49 * √2 / 4) = 21 * 28 / 2 * 49 = 6 cm.
Answer: The median Dina is 6 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.