A point belonging to the hypotenuse of a right triangle is equidistant from both legs. The distance from this point

| January 27, 2021 | education

A point belonging to the hypotenuse of a right triangle is equidistant from both legs. The distance from this point to the vertex of the right angle of the triangle is 4 roots of 2, and the lengths of the legs are 2: 3. Find the length of the larger leg

Since, by condition, KН = MН, then ВKНM is a square, Then ВН = KН = MН = ВM = ВН * Sin45 = 4 * √2 * √2 / 2 = 4 cm.

Rectangular triangles ACН and СMН are similar in acute angle, then:

AK / KН = MН / MС.

AK * CM = 4 * 4 = 16. (1).

AB / BC = 3/2.

AB = AK + 4.

BC = CM + 4.

(AK + 4) / (CM + 4) = 3/2.

3 * CM + 12 = 2 * AK + 8.

2 * AK – 3 * CM = 4. (2)

Let’s solve the system of equations 1 and 2.

AK = 16 / CM.

32 / CM – 3 * CM = 4.

(32 – 3 * CM ^ 2) = 4 * CM.

3 * CM ^ 2 + 4 * CM – 32 = 0.

Let’s solve the quadratic equation:

CM = 8/3 cm.

Then AK = 16 * 3/8 = 6 cm, AB = 6 + 4 = 10 cm.

Answer: The length of the larger leg is 10 cm.



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