# Max Area of Rectangle within Isosceles Triangle

• Izzhov
In summary, the problem is to find the maximum area of a rectangle within the boundaries of an isosceles triangle with a base of 10 and congruent sides' lengths of 13. The maximum area is proven to be 30 if one of the sides of the rectangle coincides with one of the sides of the isosceles triangle. However, it is not proven that the area of the rectangle cannot be greater if none of the sides coincide with the triangle's sides. It is noted that the area of a rectangle is the same as the area of a parallelogram with the same base and height, and the maximum area can be achieved when the parallelogram's triangles line up with two of the triangles of the
Izzhov
The Problem:
To find the maximum area of a rectangle within the boundaries of an isosceles triangle with a base of 10 and congruent sides' lengths of 13.

How Far I Was Able to Get:
I was able to prove that the max area was 30 if one of the sides of the rectangle coincides with one of the sides of the isosceles triangle. However, I was not able to prove that the area of the rectangle could not be greater if none of the sides of the rectangle coincide with any of the sides of the isosceles triangle (meaning that one or more of the vertices of the rectangle is not touching any of the edges of the triangle).

The area of a rectangle is the same as the area of a parallelogram with the same base and height.

Think about the maximum area of parallelogram you can fit into the triangle.

Hmm... I think I get it. Since a triangle can be made out of four congruent triangles, and a parallelogram can be made out of two congruent triangles, the maximum area has to be when the triangles of the parallelogram line up with two of the triangles comprising the larger one, which is one half the area of the larger triangle. Is this correct?

## What is the definition of "Max Area of Rectangle within Isosceles Triangle"?

The Max Area of Rectangle within Isosceles Triangle refers to the largest possible area of a rectangle that can be inscribed within an isosceles triangle, with the rectangle's base on the triangle's base and its two other vertices on the triangle's sides.

## Why is this concept important in mathematics and science?

This concept is important because it allows us to find the maximum possible area within a given shape, which is a common problem in optimization and geometry. It also has applications in fields such as architecture and engineering.

## What is the formula for calculating the Max Area of Rectangle within Isosceles Triangle?

The formula for calculating the Max Area of Rectangle within Isosceles Triangle is 0.5 x base x height, where the base and height are equal to the length of the triangle's base and half of the triangle's height, respectively.

## How do you find the dimensions of the rectangle with the maximum area?

To find the dimensions of the rectangle with the maximum area, you can use the formula mentioned above and plug in the values for the base and height. This will give you the length and width of the rectangle, which can then be used to construct the shape.

## Are there any real-life examples of the Max Area of Rectangle within Isosceles Triangle?

Yes, there are many real-life examples of this concept, such as finding the maximum amount of material that can be used to make a rectangular window within a triangular frame, or maximizing the area of a garden bed within a triangular space. It can also be applied in designing structures like bridges and roofs.

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