Triangles

To find this ratio, we apply the angle bisector theorem, which states that an angle bisector of a triangle divides the opposite side into segments proportional to the other two sides. Therefore, according to the angle bisector theorem, we have:

AB/BC = AD/DC

First, calculate AD:

AC = AD + DC

10.5 = AD + 7

Solving for AD, we find:

AD = 10.5 - 7 = 3.5 cm

Now, substitute AD and DC into the angle bisector theorem equation:

AB/BC = 3.5/7

Simplifying 3.5/7 to its lowest terms gives:

AB/BC = 1/2

Therefore, the ratio AB : BC is 1 : 2.

To find the least perimeter of the rectangle formed without loss of area from the equilateral triangle, we proceed as follows:

Firstly, calculate the area of the equilateral triangle. For a triangle with side , the area is given by:

Given , substituting the value:

cm²

Next, consider a rectangle with length and breadth . The area of the rectangle is equal to the area of the triangle:

The perimeter of the rectangle is given as:

To minimize the perimeter for a given area, the rectangle should be as close to a square as possible. This implies .

Setting , we get:

Thus,

Calculating:

cm

Thus, the least perimeter becomes:

cm

The least perimeter of the rectangle is cm.