Question #90914

An isosceles triangle is to be cut from a square lamina of side 50 cm. find the altitude it's of this triangle so that it vertex will be the centroid of the remaining shaded area.

Expert's answer

**Solution:**

Since the plate has one thickness over the entire plane, we can assume that the center of gravity of the resulting figure will be the point of intersection of the lines, on the opposite sides of which, the resulting figures will be the same areas.

Fig 1

In Fig 1, the intersection of the line "b" and the line "c" will be the center of gravity of the figure.

The areas on different sides of the "c" line will be the same because we fit an isosceles triangle into the square.

It remains to calculate the "x" and "y", "y" is the height of the triangle, and the answer to our task.

Hence the area of the rectangle with sides "a" and "x" should be equal to two areas of a right-angled triangle with the legs "y" and "a / 2"

St=1/2*y*a/2 Area of a right triangle (1)

Sr=a*x Rectangle area (2)

Sr=2*St (3)

a*x=2*1/2*y*a/2 (4)

x=y/2 (5)

On the other hand we know that

x+y=a (6)

x=a-y (7)

y=2*x=2*a-2*y (8)

3*y=2*a (9)

y=2/3*a=2/3*50=33.3333 (10)

**Answer:**

Isosceles triangle height 33.3333 cm

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