Triangles are fascinating geometric shapes that appear in various applications, from architecture to engineering. However, calculating the height of a triangle on the Cartesian plane can be a bit challenging if you are not familiar with the necessary formulas. In this blog post, we will guide you through the step-by-step process of determining the height of a triangle using simple and easy-to-understand explanations.

What is the Cartesian Plane?

Before we delve into the calculation of the triangle’s height, let’s first understand what the Cartesian plane is. The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane defined by two axes – the x-axis and the y-axis. It is named after the mathematician René Descartes, who introduced the coordinate system.

  • The x-axis is the horizontal line running from left to right.
  • The y-axis is the vertical line running from bottom to top.

Understanding the Triangle

To calculate the height of a triangle on the Cartesian plane, we need to identify the points of its vertices. For the sake of simplicity, let’s consider a triangle with vertices at points A, B, and C.

Calculating the Distance

To calculate the height, we first need to determine the length of the triangle’s base. The base is the line segment between two vertices of the triangle. Let’s say the base of our triangle is the line segment between point A(x1, y1) and point B(x2, y2).

The distance between two points in the Cartesian plane can be found using the distance formula, which is derived from the Pythagorean theorem. The formula is as follows:

distance = √((x2 – x1)2 + (y2 – y1)2)

Therefore, the distance between points A and B, which is the length of the triangle’s base, is:

base = √((x2 – x1)2 + (y2 – y1)2)

Determining the Area

To calculate the height of the triangle, we need to know its area. The area of a triangle can be found using the formula:

area = 0.5 * base * height

Solving for the Height

Now that we know the area of the triangle and the length of the base, we can rearrange the area formula to solve for the height:

height = (2 * area) / base

An Example Calculation

Let’s put our newfound knowledge into practice with an example: Suppose we have a triangle with vertices A(3, 4), B(7, 8), and C(5, 4). We want to calculate the height of this triangle.

First, we find the length of the base (AB):

base = √((7 – 3)2 + (8 – 4)2)

base = √(42 + 42)

base = √(16 + 16)

base = √32

base ≈ 5.66 (rounded to two decimal places)

Next, we need to calculate the area of the triangle. For simplicity, we won’t go into the derivation of the area formula but rather provide it directly:

area = 0.5 * base * height

Since we want to solve for the height, we rearrange the formula as follows:

height = (2 * area) / base

Let’s assume the area of the triangle is 12 square units:

height = (2 * 12) / 5.66

height ≈ 4.23 (rounded to two decimal places)

Calculating the height of a triangle on the Cartesian plane involves finding the length of the base and then using the area formula with some algebraic manipulation. By following the steps outlined in this blog post, you can confidently calculate the height of any triangle on the Cartesian plane. Remember, practice makes perfect, so don’t hesitate to attempt more examples to solidify your understanding of this valuable mathematical concept.

We hope this guide has been helpful in explaining how to calculate the height of a triangle on the Cartesian plane. If you have any further questions or topics you would like us to cover, feel free to leave a comment below!

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