The height of the triangle is 12 cm and divides the midline perpendicular to it into 4, 5

| September 6, 2021 | education

The height of the triangle is 12 cm and divides the midline perpendicular to it into 4, 5 cm and 2.5 cm segments. Find the perimeter of the triangle.

Since, according to the condition, the segment KM is the middle line of the triangle ABC, it divides the height H in half, and therefore the segments KO and MO are the middle lines of the triangles ABН and СBН.

The midline of a triangle is half the length of the base of the triangle to which it is parallel.

In a right-angled triangle ABH, KO = 4.5 cm, then the base AH = 2 * KO = 2 * 4.5 = 9 cm.

Then, by the Pythagorean theorem, AB ^ 2 = BH ^ 2 + AH ^ 2 = 144 + 81 = 225. AB = 15 cm.

In a right-angled triangle СBН, KO = 2.5 cm, then the base СН = 2 * MO = 2 * 2.5 = 5 cm.

Then, by the Pythagorean theorem, CB ^ 2 = BH ^ 2 + CH ^ 2 = 144 + 25 = 169. CB = 13 cm.

The length of the middle line KM is equal to: KM = KO + MO = 4.5 + 2.5 = 7 cm.

Then the base AC = 2 * KM = 2 * 7 = 14 cm.

Let’s define the perimeter of the triangle ABC.

Ravs = AB + BC + AC = 15 + 13 + 14 = 42 cm.

Answer: The perimeter of the triangle is 42 cm.



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