`30-60-90` Triangle
- Introduction
- Defining a `30-60-90` Triangle
- `30-60-90` Triangle Sides
- `30-60-90` Triangle Rule
- Proof of a `30-60-90` Triangle Sides
- What is the Area of a `30-60-90` Triangle?
- Solved Examples
- Practice Problems
- Frequently Asked Problem
Introduction
A triangle is a polygon with three sides, three vertices, and three angles. We can classify triangles based on their sides as scalene, isosceles, or equilateral. But some triangles have unique features, earning them the title of special triangles. One such example is the `30°–60°–90°` triangle. A `30` `60` `90` triangle is a special type of triangle with a `1: 2 : 3` ratio for its side lengths. It's a right triangle with angles measuring `30°`, `60°`, and `90°`. The side opposite the `30°` angle is the shortest, the side opposite the `60°` angle is longer, and the side opposite the `90°` angle is the longest, known as the hypotenuse.
These special triangles are valuable for solving geometry and trigonometry problems. Among them, the `30-60-90` triangle stands out. This lesson will lay out its concept, formula, and properties in details.
Defining a `30-60-90` Triangle
A `30-60-90` triangle is a special type of right triangle where the angles measure `30` degrees, `60` degrees, and `90` degrees. These angles are in a unique ratio of `1:2:3` and so are the sides opposite to these angles. Triangle `ABC` and triangle `PQK` are examples of `30-60-90` triangles. In these triangles, the relationship between the lengths of their sides is always the same. This means that no matter how big or small the triangle is, the proportions of the sides remain constant.
Here, in the triangle `ABC`, `∠ C = 30°`,`∠ A = 60°`, and `∠ B = 90°` and in the triangle `PQK`, `∠ P = 30°`,`∠ K = 60°`, and `∠ Q = 90°`
`30-60-90` Triangle Sides
Now, let's look into the side lengths of a `30-60-90` triangle.
A `30-60-90` triangle is quite special because its side lengths have a consistent relationship with one another. Imagine triangle `ABC`, where angle `C` measures `30` degrees, angle A measures `60` degrees, and angle `B` is the right angle at `90` degrees. The sides of a `30-60-90` triangle always follow the ratio of `1:sqrt3:2`, which means they're always in proportion. This relationship is sometimes called the `30-60-90` triangle formula for sides: `y`, `ysqrt3`, `2y`.
Now, let's break down the sides:
- The side opposite the `30`-degree angle, `AB` (we'll call it `y`), is always the smallest because `30` degrees is the smallest angle in this triangle.
- The side opposite the `60`-degree angle, `BC` (which equals `y` times the square root of `3`), is of medium length since `60` degrees is in the middle angle in terms of its measure.
- Finally, the side opposite the `90`-degree angle, the hypotenuse `AC` (equals `2y`), is the longest because `90` degrees is the largest angle.
You can prove this using the Pythagorean theorem, but understanding this ratio is crucial for working with these triangles effectively.
`30-60-90` Triangle Rule
The `30-60-90` triangle rule helps us find the lengths of the sides in a specific type of triangle. It states that in a `30-60-90` triangle, you can determine the length of any side if you know the length of at least one side. This is often referred to as the `30-60-90` triangle rule.
Here's how it works:
- If you know the length of the base of the triangle (opposite the `30`-degree angle), denoted as ``a``, you can find the other sides.
The perpendicular (opposite the `60`-degree angle) will be ‘`asqrt3`’.
The hypotenuse (opposite the `90`-degree angle) will be ‘`2a`’.
- Similarly, if you have the length of the perpendicular (opposite the `60`-degree angle), marked as ‘`a`’, you can determine the other sides.
The base (opposite the `30`-degree angle) will be ‘`a/sqrt3`’.
The hypotenuse remains ‘`2a`’.
- Lastly, if you're given the length of the hypotenuse, let's say it's '`a`', then you can find the other sides.
The base will be '`(sqrt3a)/2`'.
The perpendicular will be ‘`a/2`’.
Remembering these relationships simplifies finding the sides of a `30-60-90` triangle.
Proof of a `30-60-90` Triangle Sides
Let's explore the proof of the `30-60-90` triangle theorem. Start with an equilateral triangle `ABC`, where all sides have the same length '`a`'.
Now, draw a perpendicular line from vertex `A` to side `BC`, creating point `D`. This perpendicular line bisects side `BC`, dividing it into two equal parts.
Now, focus on triangles `ABD` and `ADC`. Both are `30-60-90` triangles, meaning they have angles measuring `30°`, `60°` and `90°`. Since they share the same angle measures, they are similar and right-angled triangles. This similarity allows us to apply the Pythagorean theorem to find the length of `AD`.
Using the Pythagorean theorem on triangle `ABD`,
`(AB)^2 = (AD)^2 + (BD)^2`
Substituting the known values we get,
`a^2 = (AD)^2 + (a/2)^2`
Simplifying this, we find `(AD)^2 = (3a^2)/4`, which leads to `AD = (asqrt3)/2`.
We also know that `BD = a/2` and `AB = a`. These side lengths follow a specific ratio of `a/2 : (asqrt3)/2 : a`. Simplifying this ratio, we get `1 : sqrt3 : 2`. This ratio, derived from our equilateral triangle, is the essence of the `30-60-90` triangle theorem.
What is the Area of a `30-60-90` Triangle?
To calculate the area of any triangle, we use the formula: area `= (1/2) × \text{base} × \text{height}`.
In a right-angled triangle, the height is the perpendicular from the base to the vertex opposite the right angle. So, the formula for the area of a right-angled triangle becomes area `= (1/2) × "base" × "perpendicular"`.
Now, focusing on the `30-60-90` triangle, let's assume its base is labeled as '`a`', and the hypotenuse (the longest side) is labeled as '`AC`'. We've already learned how to find the hypotenuse when the base is given.
Using the formula for the area of a triangle, we find that the perpendicular of the triangle equals `a/sqrt3`.
Substituting the values into the area formula, we get: area `= (1/2) × a × (a/sqrt3)`.
Simplifying, we find the area of the `30-60-90` triangle when the base (the side of medium length) is given as '`a`' is: `a^2 / (2sqrt3)`.
Solved Examples
Example `1`: The shortest side of a `30-60-90` triangle is `6` cm. Find the length of the hypotenuse (longest side) and the other leg.
Solution:
Since the ratio of the sides in a `30-60-90` triangle is \(1: \sqrt{3}: 2\), and we're given one leg as `6` cm, we can find the other sides:
Length of the hypotenuse `=` \(2 \times 6 = 12\) cm
Length of the other leg `=` \(\sqrt{3} \times 6 = 6\sqrt{3}\) cm
Example `2`: The length of the hypotenuse of a `30-60-90` triangle is `10` in. Find the length of the other two sides.
Solution:
Since the ratio of the sides in a `30-60-90` triangle is \(1 : \sqrt{3} : 2\), and we're given the hypotenuse as `10` in, we can find the other sides:
Length of the shorter leg `=` \(\dfrac{1}{2} \times 10 = 5\) in
Length of the other leg `=` \(\sqrt{3} \times 5 = 5\sqrt{3}\) in
Example `3`: The triangle has sides measuring \(3\sqrt{2}\), \(3\sqrt{6}\), and \(3\sqrt{8}\). Find the angles of this triangle.
Solution:
Dividing each side by \(3\sqrt{2}\), we get \(1\), \(\sqrt{3}\), and \(2\).
Since these ratios match the `30-60-90` triangle pattern `(1: sqrt3: 2)`, the given triangle indeed follows the `30-60-90` triangle rule.
Thus, the angles of the triangle are \(30°\), \(60°\), and \(90°\).
Example `4`: The length of one leg of a `30-60-90` triangle is `6` cm. Find the area of the triangle.
Solution:
Since the area of a triangle can be calculated using the formula \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\), and we know the length of one leg (base) is `6` cm, we need to find the height (perpendicular).
In a `30-60-90` triangle, the relationship between the sides is \(1 : \sqrt{3} : 2\). So, if one leg is `6` cm, the perpendicular will be `\frac{6}{\sqrt{3}}` cm.
Now, substituting the values into the area formula:
\(Area = \frac{1}{2} \times 6 \times {6}{\sqrt{3}}\)
\(= \frac{1}{2} \times 36{\sqrt{3}}\)
\(= 18{\sqrt{3}}\)
So, the area of the triangle is \(18\sqrt{3} \, \text{cm}^2\).
Example `5`: A ramp is placed with one end at ground level and the other end raised `5` feet above the ground level. The angle of inclination between the ramp and the ground is `30` degrees. Find the length of the ramp.
Solution:
Since the ramp forms a right triangle with the ground, and the angle of inclination is `30` degrees, it indicates a `30-60-90` triangle.
In a `30-60-90` triangle, the relationship between the sides is \(1 : \sqrt{3} : 2\). Given that one leg (opposite the `30`-degree angle) is `5` feet, we can find the length of the hypotenuse (length of the ramp).
So, the length of the hypotenuse (ramp) is \(2 \times 5 = 10\) feet.
Therefore, the length of the ramp is `10` feet.
Practice Problems
Q`1`: Given that one leg of a `30-60-90` triangle measures \(4\sqrt{3}\) units, find the length of the hypotenuse of the triangle.
- \(8\) units
- \(8\sqrt{3}\) units
- \(8\sqrt{2}\) units
- \(16\) units
Answer: b
Q`2`: The hypotenuse of a `30-60-90` triangle is \(12\) meters. What is the length of the shorter leg of the triangle?
- \(4\) meters
- \(4\sqrt{3}\) meters
- \(6\) meters
- \(6\sqrt{3}\) meters
Answer: c
Q`3`: Given that one leg of a `30-60-90` triangle measures \(10\) inches, find the length of the longer leg of the triangle.
- \(5\) inches
- \(5\sqrt{3}\) inches
- \(10\) inches
- \(10\sqrt{3}\) inches
Answer: b
Q`4`: A ladder is placed against a wall at an inclination of `60°` with respect to the ground. The foot of the ladder is `6` meters away from the wall. The ladder reaches a point `4.5` meters above the ground. What is the length of the ladder?
- `7` meters
- `7.5` meters
- `8` meters
- `10.25` meters
Answer: b
Q`5`: A slide in a playground has an incline of `30` degrees with the ground. If the bottom of the slide is `2` meters away from the base of the ladder, and the top of the slide is `1` meter above the base of the ladder, what is the length of the slide? Give your answer rounded to `2` decimal places.
- `3.50` meters
- `4.15` meters
- `2.23` meters
- `5.72` meters
Answer: C
Frequently Asked Questions
Q`1`: What Is the perimeter of a `30-60-90` Triangle?
Answer: The perimeter of a `30-60-90` triangle, where the smallest side measures '`a`', encompasses the sum of all three sides. The remaining two sides measure '`asqrt3`' and '`2a`'. Therefore, the triangle's perimeter equals '`a + asqrt3 + 2a`', which simplifies to '`3a + asqrt3`'. This can be further expressed as '`asqrt3(1 + sqrt3)`'.
Q`2`: How is the `30-60-90` triangle similar to `45-45-90`?
Answer: The `30-60-90` triangle and the `45-45-90` triangle share several similarities.
- They are both right triangles.
- They both adhere to the Pythagorean theorem.
- The total of their internal angles equals `180` degrees.
Q`3`: Which side of a `30-60-90` triangle is the longer leg?
Answer: The longer leg of a `30-60-90` triangle is the side that exceeds the length of the shorter leg but falls short of the hypotenuse. It corresponds to the side positioned opposite the `60`-degree angle.
Q`4`: What are the side ratios in a `30-60-90` triangle?
Answer: The side ratios in a `30-60-90` triangle are `1: sqrt3: 2`. This means that the ratio of the lengths of the sides opposite the `30°` angle, `60°` angle, and `90°` angle respectively follows this pattern.
Q`5`: What are some common misconceptions about `30-60-90` triangles?
Answer: One common misconception is assuming that all right triangles are `30-60-90` triangles. While `30-60-90` triangles are a subset of right triangles, not all right triangles follow the same angle and side ratios as a `30-60-90` triangle.