Finding the Area of a Triangle with Given Sides and Median

In this article, we explore the process of finding the area of a triangle when given the lengths of its sides and the length of a median. The example we will work through is a triangle ABC with sides AB 10, AC 21, and median AM 8.5. This problem requires the use of specific formulas and mathematical steps to determine the area.

Understanding the Problem

We start with the given information: triangle ABC with AB 10, AC 21, and median AM 8.5. The goal is to find the area of the triangle. The method we will use involves the relationship between the median and the sides of the triangle.

Step 1: Determine the Length of Side BC

First, we need to express the length of side BC (denoted as a) using the formula for the median in a triangle. The formula for the length of a median from vertex A to side BC is:

m_a frac{1}{2} sqrt{2b^2 2c^2 a^2}

Substituting the known values:

8.5 frac{1}{2} sqrt{221^2 210^2 a^2}

Calculating 212 and 102:

212 441, 102 100

Substitute these values into the formula:

8.5 frac{1}{2} sqrt{2441 2100 a^2}

Calculate 2441 - 2100:

2441 - 2100 341

Now we have:

8.5 frac{1}{2} sqrt{341 - a^2}

Multiplying both sides by 2:

17 sqrt{341 - a^2}

Squaring both sides:

289 341 - a^2

Rearranging:

a2 341 - 289 52

a sqrt{52}

Step 2: Calculate the Semi-Perimeter sm

The semi-perimeter sm of the triangle is:

sm frac{a b c}{2} frac{sqrt{52} 21 10}{2} frac{sqrt{52} 31}{2}

Step 3: Determine the Area Using the Formula

Finally, we use the formula for the area of a triangle in terms of its sides and a median:

Area frac{4}{3} sqrt{sm (sm a) (sm b) (sm c)}

Calculating:

sm - a frac{sqrt{52} 31}{2} - sqrt{52} frac{31 - sqrt{52}}{2}

sm - b frac{sqrt{52} 31}{2} - 21 frac{sqrt{52} 31 - 42}{2} frac{sqrt{52} - 11}{2}

sm - c frac{sqrt{52} 31}{2} - 10 frac{sqrt{52} 31 - 20}{2} frac{sqrt{52} 11}{2}

Substitute these values into the area formula:

Area frac{4}{3} sqrt{left(frac{sqrt{52} 31}{2}right) left(frac{31 - sqrt{52}}{2}right) left(frac{sqrt{52} - 11}{2}right) left(frac{sqrt{52} 11}{2}right)}

This will simplify to:

Area approx 84 square units

Final Calculation

The area of triangle ABC is approximately 84 square units.

Conclusion

This method demonstrates the application of mathematical formulas and algebraic manipulation to find the area of a triangle given the lengths of its sides and the length of a median. The problem involves steps from geometry and algebra, leading to the use of Heron's formula and specific median relations to determine the area. This technique can be applied to other similar problems in geometry.