In a triangle MHP, the side MH is no longer than 12, the side of HP is no longer than 5, and its area is at least 30.

| May 20, 2021 | education

In a triangle MHP, the side MH is no longer than 12, the side of HP is no longer than 5, and its area is at least 30. Find the diameter of the circle around the triangle MHP.

The area of a triangle is equal to half the product of the lengths of its sides by the sine of the angle between them.

Smnr = МН * НР * SinМНР / 2.

Let МН = 12 cm, РН = 5 cm, then Smnr = 12 * 5 * SinМНР / 2 = 30 * SinМНР

By the condition Smnr ≥ 30 cm2, then SinMHP cannot be <1.

Then SinМНР = 1, and the angle РНМ = 90, which means the triangle МНР is rectangular.

By the Pythagorean theorem PM ^ 2 = PH ^ 2 + MH ^ 2 = 25 + 144 = 169.

PM = 13 cm.

Since the circle is circumscribed about a right-angled triangle, its diameter coincides with the hypotenuse of the triangle.

Answer: The diameter of the circumscribed circle is 13 cm.



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