Inscribe a Square in a Right Triangle with Sides 5, 12, and √97: Calculation and Application

In this article, we will delve into the process of inscribing a square within a triangle whose sides are 5, 12, and √97. We will start by determining the area of the triangle using Heron's formula, followed by the calculation of the inscribed square's area.

Step 1: Calculate the Area of the Triangle Using Heron's Formula

To find the area of the triangle with sides 5, 12, and √97, we first need to calculate the semi-perimeter (s).

The semi-perimeter is given by:

s (frac{a b c}{2}) where a 5, b 12, c (sqrt{97})

So, the semi-perimeter is:

s (frac{5 12 sqrt{97}}{2} frac{17 sqrt{97}}{2})

Now, we can use Heron's formula to find the area of the triangle:

Area (sqrt{s(s - a)(s - b)(s - c)})

Substituting the values, we get:

s - a (frac{17 sqrt{97}}{2} - 5 frac{7 sqrt{97}}{2})

s - b (frac{17 sqrt{97}}{2} - 12 frac{-7 sqrt{97}}{2})

s - c (frac{17 sqrt{97}}{2} - sqrt{97} frac{17 - sqrt{97}}{2})

Thus, the area of the triangle is:

Area (sqrt{(frac{17 sqrt{97}}{2})(frac{7 sqrt{97}}{2})(frac{-7 sqrt{97}}{2})(frac{17 - sqrt{97}}{2})})

Calculating this value, we find the exact area of the triangle to be 24.

Step 2: Calculate the Area of the Inscribed Square

The area of a square inscribed in a triangle can be calculated using the formula:

A (frac{abc}{a b c^2})

Where a 5, b 12, c (sqrt{97})

Let's calculate the components of the formula:

abc 5 (times) 12 (times) (sqrt{97}) 60(sqrt{97})

a b c 5 12 (sqrt{97}) 17 (sqrt{97})

a b c^2 17 (sqrt{97}) 97 386 34(sqrt{97})

Substituting these values into the formula:

A (frac{60sqrt{97}}{386 34sqrt{97}})

This gives us the exact area of the inscribed square within the triangle.

Conclusion

The area of the square inscribed in the triangle is:

A (frac{60sqrt{97}}{386 34sqrt{97}})

This expression can be simplified further if numerical values are needed, but this is the exact form of the area.

For a more intuitive understanding, consider a right triangle with sides 5, 12, and √97. The area of the triangle can be determined quickly using Heron's formula, and the height of the triangle can be calculated as 4. If we inscribe the square within the triangle, the height of the triangle will be divided proportionally.

The area of the green triangle is 1/2 (times) 4 (times) x, and the combined pink area is 1/2 (times) 12 - x (times) x. Therefore:

1/2 (times) 4 (times) x 1/2 (times) 12 - x (times) x 25

Solving this equation, we find x 3.

Thus, the area of the inscribed square within the triangle is a direct result of the proportional division of the triangle's height and can be mathematically derived using the methods outlined above.