Question

Find the area of the triangle with vertices on the coordinate axes at the points \((a, 0,0),(0, b, 0),\) and \((0,0, c)\) in terms of \(a, b,\) and \(c\)

Short answer

Question: Find the area of a triangle with vertices on the coordinate axes at the points (a, 0, 0), (0, b, 0), and (0, 0, c) in terms of a, b, and c. Answer: The area of the triangle is given by the following expression: \(\sqrt{\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{a^2+b^2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{b^2+c^2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{a^2+c^2}\right)}\)

Step-by-step solution

Find the lengths of the sides

To find the lengths of the sides, we can use the distance formula in three-dimensional space: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\). Using the distance formula, we can find the length of the sides AB, BC, and AC, where A=(a,0,0), B=(0,b,0) and C=(0,0,c). For side AB, let A=(a,0,0) and B=(0,b,0): \(AB=\sqrt{(0-a)^2+(b-0)^2+(0-0)^2}=\sqrt{a^2+b^2}\). For side BC, let B=(0,b,0) and C=(0,0,c): \(BC=\sqrt{(0-0)^2+(0-b)^2+(c-0)^2}=\sqrt{b^2+c^2}\). For side AC, let A=(a,0,0) and C=(0,0,c): \(AC=\sqrt{(0-a)^2+(0-0)^2+(c-0)^2}=\sqrt{a^2+c^2}\). So, the lengths of the sides are \(AB=\sqrt{a^2+b^2}\), \(BC=\sqrt{b^2+c^2}\), and \(AC=\sqrt{a^2+c^2}\).

Calculate the semi-perimeter

To apply Heron's formula, we first need to calculate the semi-perimeter (s) of the triangle. This is done by taking half the sum of the lengths of all three sides: \(s = \frac{AB + BC + AC}{2} = \frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}\).

Apply Heron's formula to find the area

Now that we have the semi-perimeter, we can apply Heron's formula to find the area of the triangle. Heron's formula is given as follows: \(Area = \sqrt{s(s-AB)(s-BC)(s-AC)}\) Using our values for s, AB, BC, and AC, we have: \(Area = \sqrt{\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{a^2+b^2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{b^2+c^2}\right)\left(\frac{\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{a^2+c^2}}{2}-\sqrt{a^2+c^2}\right)}\) This expression represents the area of the triangle with vertices on the coordinate axes at the points \((a, 0,0),(0, b, 0),\) and \((0,0, c)\) in terms of \(a, b,\) and \(c\).

Key concepts

Distance Formula

In geometry, the distance formula allows us to determine the distance between two points in space. When working in three-dimensional space, this formula expands to accommodate the extra dimension, ensuring accuracy in calculations. If you have points

• Point 1: \(x_1, y_1, z_1\)
• Point 2: \(x_2, y_2, z_2\)

The distance between them can be calculated using:
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]This powerful formula is the foundation for solving various problems in 3D geometry.

• For side \(AB\) between points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), this formula will help you find the length accurately.
• Substitute the coordinates into the formula as appropriate for each side to find its length. For example, substitute for \(AB\) as \(A(a, 0, 0)\) and \(B(0, b, 0)\) into the formula gives: \[AB = \sqrt{a^2 + b^2} \].

Heron's Formula

When it comes to calculating the area of a triangle, Heron's formula emerges as a fantastic tool, especially when all three side lengths are known. This formula is particularly useful for triangles in any dimension, including those defined by points in a 3D coordinate system.
Heron's formula is expressed as:

• First, find the semi-perimeter \(s\) of the triangle.
• Next, use the formula for the area:\[Area = \sqrt{s(s-a)(s-b)(s-c)}\]where \(a, b,\) and \(c\) are the lengths of the sides of the triangle.

Heron's formula simplifies the process by not requiring the height of the triangle. Just ensure that the values calculated from the distance formula are used accurately to find the semi-perimeter and ultimately the area.

Semi-Perimeter

The concept of the semi-perimeter is integral when utilizing Heron's formula to find the area of a triangle. Essentially, the semi-perimeter is half the sum of the lengths of the triangle's sides. It acts as a middle step, setting up for the efficient calculation of the area through Heron's approach.

• For a triangle with side lengths \(AB, BC,\) and \(AC\), the semi-perimeter \(s\) is given by:\[s = \frac{AB + BC + AC}{2}\]
• For our specific problem, this would translate to:\[s = \frac{\sqrt{a^2+b^2} + \sqrt{b^2+c^2} + \sqrt{a^2+c^2}}{2}\]

Calculating the semi-perimeter neatly organizes our data, making the application of Heron's formula straightforward. This means less hassle in finding your required area by reducing potential mistakes from misplaced operations.