Use Theorem 11.4.1 to show that the area of an equilateral...

Key Concepts

Triangle Area Formula

The area of a triangle is determined by the formula one-half times the base times the height. This formula is fundamental in geometry and serves as the starting point for calculating the area of more specific types of triangles.

Equilateral Triangle Properties

An equilateral triangle is characterized by three congruent sides and three equal angles, each measuring 60 degrees. These properties introduce symmetry into the triangle, simplifying the calculations of other measurements, such as its altitude.

Altitude of an Equilateral Triangle

The altitude (or height) of an equilateral triangle can be deduced by drawing a perpendicular line from one vertex to the opposite side. This divides the triangle into two 30-60-90 right triangles, allowing the height to be expressed in terms of the side length using the ratio inherent to 30-60-90 triangles.

Application of Geometric Theorems

Using formal results, like Theorem 11.4.1, reinforces the rigor in deriving geometric formulas. Such theorems provide a systematic approach to confirming that the area of an equilateral triangle with side length s is indeed (s²?3)/4 by connecting established geometric properties with the general area formula.

Transcript

00:01 In the given question, we are told to show that the area of an equilateral triangle with sides of length given as s units is found as a is equal to s square divided by 4 times the square root of 3.

00:44 So this is what we have to show over here.

00:47 So what we could do is we could just take an equilateral triangle.

00:52 Right.

00:53 So first we take an equilateral triangle.

00:57 The length of sides of this equilateral triangle are s.

01:02 And now what we generally use to find the area of a triangle is the formula that is given by half of the base length of the triangle times the height of the triangle.

01:19 Right.

01:20 So now what we could do is over here we do know the base length but we don't know what is the height of this triangle.

01:32 Right.

01:32 But we do know a property that is the altitude of an equilateral triangle would bisect this side.

01:44 So this means when you draw the altitude which we can denote as the height h, it divides this base into two equal parts that is s by 2 and since this altitude will be perpendicular to this base what we could now do is we can just say let's say this is the triangle abc.

02:08 This altitude is bisecting the side at the point d.

02:14 Then you can take the right angled triangle adc with height h and side lengths s by 2 and s and we can see that s over here would be giving us the length of the hypotenuse.

02:30 Right.

02:31 So in the next step what we could do is we could make use of a theorem that says which is called the pythagoras theorem...

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