Calculate the perimeters of the triangles into which the right-angled triangle ABC

| September 23, 2021 | education

Calculate the perimeters of the triangles into which the right-angled triangle ABC is divided by the height drawn from the vertex of angle C, if the projections of the legs BC and AC on the hypotenuse are 36 cm and 64 cm, respectively

Let’s use the property of the height of a right-angled triangle.

The height of a right-angled triangle is equal to the square root of the product of the lengths of the segments by which the height divides the hypotenuse.

CH = √ (AH * BH) = √36 * 64 = √2304 = 48 cm.

Consider a right-angled triangle ACH, and by the Pythagorean theorem we define the hypotenuse AC.

AC ^ 2 = CH ^ 2 + AH ^ 2 = 48 ^ 2 + 64 ^ 2 = 2304 + 4096 = 6400.

AC = √6400 = 80 cm.

CH = √ (AH * BN) = √36 * 64 = √2304 = 48 cm.

Consider a right-angled triangle ВСH, and by the Pythagorean theorem we define the hypotenuse ВС.

BC ^ 2 = CH ^ 2 + BH ^ 2 = 48 ^ 2 + 36 ^ 2 = 2304 + 1296 = 3600.

BC = √3600 = 60 cm.

Determine the perimeter of the BCH triangle.

Рвсн = ВС + СН + ВН = 60 + 48 + 36 = 144 cm.

Determine the perimeter of the triangle ACH.

Rasn = AC + CH + AH = 80 + 48 + 64 = 192 cm.

Answer: Rvsn = 144 cm, Rasn = 192 cm.



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