Triangles are one of the most basic shapes in geometry and are used in a wide variety of mathematical applications. A particularly important triangle type is the 30-60-90 triangle, which has some unique properties that make it useful in a number of mathematical contexts.
What is the 30-60-90 triangle?
A 30-60-90 triangle is a special type of right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. This means that one of the angles is half of the right angle while the other is one third of the right angle. The 30-60-90 triangle is an isosceles triangle, meaning that two sides are equal in length.
The sides of a 30-60-90 triangle have a specific ratio that is useful in a variety of mathematical applications. This ratio is based on the length of the hypotenuse, the longest side of the triangle opposite the right angle. The other two sides are referred to as adjacent and opposite sides.
- The ratio of the sides in a 30-60-90 triangle is as follows:
- The side opposite the 30 degree angle is half the length of the hypotenuse.
- The side opposite the 60 degree angle is √3 times the length of the side opposite the 30 degree angle.
- The hypotenuse is twice the length of the side opposite the 30 degree angle.
That is, if the length of the hypotenuse is known, the lengths of the other two sides can be easily calculated using this ratio. For example, if the hypotenuse is 6 units long, the side opposite the 30 degree angle would be 3 units long and the side opposite the 60 degree angle would be 3√3 units long.
Here, in triangle ABC,∠C = 30°,∠A = 60° and∠B = 90° and in triangle PQK,∠P = 30°,∠K = 60° and∠Q = 90°
What is the 30-60-90 triangle formula?
The 30-60-90 formula is useful in a variety of mathematical contexts, including trigonometry, geometry, and physics. For example, it can be used to calculate the height of an object that is a known distance from an observer, or the distance between two objects that are at a known height.
The 30-60-90 formula is a mathematical rule that applies to special right triangles whose angles are in the ratio 1:2:√3. Here are some of its characteristics or specifications:
- The angles in a 30-60-90 triangle are 30 degrees, 60 degrees, and 90 degrees.
- The sides of a 30-60-90 triangle are in the ratio 1:√3:2.
- The hypotenuse of a 30-60-90 triangle is twice the length of the shorter leg.
- The longer leg of a 30-60-90 triangle is √3 times the length of the shorter leg.
- The area of a 30-60-90 triangle can be calculated by taking half the product of the two shorter sides.
- The perimeter of a 30-60-90 triangle is equal to the sum of the lengths of all three sides.
The 30-60-90 formula can be used to find the missing side lengths or angles of a triangle when at least one side length or angle is known.
Besides its mathematical applications, the 30-60-90 triangle also has some interesting properties from a visual point of view. If you draw a 30-60-90 triangle inside a circle, the side opposite the 30 degree angle is the diameter of the circle, while the side opposite the 60 degree angle is the radius of the circle.
In summary, the 30-60-90 formula is an important mathematical tool that has a wide range of applications in geometry, trigonometry, and physics. Its simple aspect ratio makes it easy to use and remember, and its unique properties make it a fascinating object to study from a visual perspective.
30-60-90 triangle set:
The 30-60-90 triangle theorem states that
In a triangle with angles of 30, 60, and 90 degrees, the hypotenuse is twice the length of the shortest side. Also, the length of the other side is equal to the square root of three times the length of the shortest side.
30-60-90 Dreiecksnachweis:
Suppose we have an equilateral triangle ABC in which all three sides have the same length "a".
Next, draw a perpendicular line from vertex A to side BC, intersecting at point D on side BC of equilateral triangle ABC. It is worth noting that in an equilateral triangle, the perpendicular line bisects the opposite side.
Triangle ABD and ADC are two right triangles with angles of 30, 60 and 90 degrees. Since both triangles are similar, we can use Pythagorean theorem to find the length of AD.
Using the Pythagorean theorem we get:
(AB)² = (AD)² + (BD)²
a² = (AD)² + (a/2)²
a² – (a/2)² = (AD)²
3a²/4 = (AD)²
Taking the square root of both sides, we get:
AD = (a√3)/2
BD = a/2
AB = a
These side lengths follow the same ratio of a/2 : (a√3)/2 : a, which simplifies to 1 : √3 : 2.
So if we multiply the ratio by 2 and divide by "a", we get:
(2a)/(2a) : (2a√3)/(2a) : (2a/a)
This simplifies to 1:√3:2, which is known as the 30-60-90 set of triangles.
30-60-90 triangle rule:
The 30-60-90 triangle rule allows us to determine the measure of each of the three sides of a 30-60-90 triangle by knowing the measure of at least one side. The following table illustrates how to use the 30-60-90 triangle rule to determine the sides of a 30-60-90 triangle:
Area of a 30-60-90 triangle
The formula for calculating the area of a triangle is given by (1/2) × base × height. In a right triangle, altitude is the length of the perpendicular from the vertex of the right angle to the hypotenuse. As a result, the formula for calculating the area of a right triangle can be written as (1/2) × base × perpendicular.
Now let's apply this formula to find the area of a 30-60-90 triangle.
Assuming that the base BC of the triangle ABC is equal to "a" and the hypotenuse of the triangle is denoted by AC, we can use the formula we learned earlier to find the length of the perpendicular AD.
Recall that the length of the hypotenuse AC is given by 2a, where "a" is the length of the shortest side of the triangle.
Therefore the length of the perpendicular AD is:
AD = (a√3)/2
To find the area of the 30-60-90 triangle we can use the formula for the area of a triangle given by:
Area = (1/2) × base area × height
In this case, the base is "a" and the height is AD, which is (a√3)/2.
Thus, the area of the 30-60-90 triangle when the base (side of mean length) is "a" is:
Area = (1/2) × a × (a√3)/2
Area = a²/(2√3)
Therefore, when the length of the base is given as "a", the area of the 30-60-90 triangle is equal to one square divided by twice the square root of three (a²/(2√3)).
example 1
A ladder leans against a wall. The base of the ladder is 6 feet from the wall and the top of the ladder is 8 feet from the floor. How long is the ladder?
Solution:
We can model the situation as a right triangle, with the wall and floor forming the legs of the triangle and the ladder forming the hypotenuse.
Let c be the length of the ladder. Then we have:
a = 6 (the distance from the wall to the base of the ladder) b = 8 (the height of the ladder from the floor)
Using the Pythagorean theorem we have:
c^2 = a^2 + b^2 c^2 = 6^2 + 8^2 c^2 = 36 + 64 c^2 = 100 c = 10
Therefore, the length of the ladder is 10 feet.
example 2
We get a triangle with sides 2√2, 2√6, and 2√8. We want to find the angles of this triangle.
Solution:
We can use the law of cosines to find the angles of the triangle. The law of cosines states that in a triangle with sides a, b and c, the cosine of one of the angles, say A, is given by:
cos(A) = (b^2 + c^2 – a^2) / (2bc)
Similarly, we can find the cosines of the other two angles, say B and C, using the same formula with the appropriate sides.
With this formula we have:
cos(A) = (2√6)^2 + (2√8)^2 – (2√2)^2 / (2 * 2√6 * 2√8) cos(A) = 24 + 32 – 8 / (4√12) cos(A) = 48 / (4√12) cos(A) = 4√3 / 9
Similarly, we can find the cosine values of angles B and C:
cos(B) = (2√2)^2 + (2√8)^2 – (2√6)^2 / (2 * 2√2 * 2√8) cos(B) = 8 + 32 – 24 / (4√16) cos(B) = 16 / (4 * 2) cos(B) = 1/2
cos(C) = (2√2)^2 + (2√6)^2 – (2√8)^2 / (2 * 2√2 * 2√6) cos(C) = 8 + 24 – 32 / (4√12) cos(C) = 0
Now that we have the cosines of the three angles, we can use the inverse cosine function to find the angles themselves:
A = cos^-1(4√3 / 9) ≈ 38,24° B = cos^-1(1/2) = 60° C = cos^-1(0) = 90°
Therefore the angles of the triangle are approximately 38.24°, 60° and 90°.
Example 3
We get a triangle with sides of 4 units, √48 units, and 8 units. We want to check if this triangle is a 30-60-90 triangle.
Remember that a 30-60-90 triangle is a right triangle with angles of 30°, 60°, and 90° and side lengths in the ratio 1:√3:2.
Solution:
To check if the given triangle is a 30-60-90 triangle, we need to check if its side lengths are in the ratio 1:√3:2.
Let's start by dividing the longest side, 8 units, by the middle side, √48 units:
8 / √48 = 8 / (4√3) = 2 / √3
Now let's divide the median side, √48 units, by the shortest side, 4 units:
√48 / 4 = √(16*3) / 4 = 2√3 / 2 = √3
So the ratio of the side lengths is:
4 : √48 : 8 = 1 : √3 : 2
Since the ratio of the side lengths is in the form 1:√3:2, which is the ratio for a 30-60-90 triangle, we can conclude that the given triangle is in fact a 30-60-90 triangle.
So the angles of the triangle are 30°, 60° and 90°, and the lengths of the sides are in the ratio 1:√3:2.