The lengths of the sides of the triangle ABC are respectively equal: BC = 15, AB = 13, AC = 4.

| June 25, 2021 | education

The lengths of the sides of the triangle ABC are respectively equal: BC = 15, AB = 13, AC = 4. Plane a, passing through the AC side, forms an angle of 30 degrees with the plane of the triangle. Find the distance from vertex B to plane a.

Let us determine the area of the triangle ABC by Heron’s theorem.

The semi-perimeter of the triangle ABC is equal to: p = (AB + BC + AC) / 2 = (13 + 15 + 4) / 2 = 16 cm.

Then Sav = √16 * (16 – 15) * (16 – 13) * (16 – 4) = √16 * 1 * 3 * 12 = √576 = 24 cm2.

Let’s build the height BH of the triangle ABC.

Then Savs = AC * BH / 2 = 24 cm2.

BH = 2 * 24/4 = 12 cm.

The distance from the vertex B to the plane is the perpendicular BH, then the triangle BHK is rectangular, in which the angle BHK = 30, then BK = BH / 2 = 12/2 = 6 cm.

Answer: From the top B to the plane 6 cm.



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