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Last modified on August 3rd, 2023

A 30-60-90 triangle is a special right triangle whose three angles are 30°, 60°, and 90°. The triangle is special because its side lengths are in the ratio of 1: √3: 2 (x: x√3: 2x for shorter side: longer side: hypotenuse).

Since a 30-60-90 triangle is a right triangle, the Pythagoras formula a^{2} + b^{2} = c^{2}, where a = longer side, b = shorter side, and c = hypotenuse is also applicable. For example, the hypotenuse can be obtained when the two other sides are known as shown below.

⇒ c^{2 }= x^{2 }+ (x√3)^{2}

⇒ c^{2 }= x^{2 }+ (x√3) (x√3)

⇒ c^{2} = x^{2 }+ 3x^{2}

⇒ c^{2} = 4x^{2}

Squaring both sides, we get,

√c^{2 }= √4x^{2}

c = 2x

- The angles are in the ratio 1: 2: 3, which are in arithmetic progression
- The sides are in the ratio 1: √3: 2 (x: √3x: 2x)
- The side opposite the 30° angle is the shorter side, denote by x
- The side opposite the 60° angle is the longer side, denote by x√3
- The side opposite the 90° angle is the hypotenuse, denote by 2x
- All 30-60-90 triangles are similar triangles
- Two 30-60-90 triangles sharing the longer side form an equilateral triangle

From the above properties, we get some basic rules applicable in all 30-60-90 triangles. The three side lengths are always in the ratio of 1: √3: 2 and the shortest side is always the smallest angle (30°), while the longest side is always opposite the largest angles (90°). These rules are useful for solving the 30-60-90 theorem that we will deal with in the next section.

Thus, the properties 2, 3, 4, and 5 are collectively called the 30-60-90 triangle theorem, which is summarized below:

- The hypotenuse is twice the length of the short leg
- The length of the longer side is √3 times the shorter side

To prove 30-60-90 Triangle Theorem

To prove:

Let △ABC be an equilateral triangle with each side length equal to x.

Proof:

Given: △ABC is an equilateral triangle with side length ‘x’

Proof:

A perpendicular line is drawn from vertex A to side BC that meets at point D such that it bisects the side BC. The two triangles formed, △ABD and △ADC are similar.

Since, both the triangles are right triangles; here we will use the Pythagorean Theorem to find the length of AD.

Thus,

AB^{2 }= AD^{2} + BD^{2}

x^{2} = AD^{2 }+(x/2)^{2}

x^{2} – (x/2)^{2} = AD^{2}

AD^{2} = 3x^{2}/4 =x√3/2

BD = x/2

AB = x

Thus, the three sides are in the ratio of:

x/2: x√3/2: x

Multiplying by 2, we get,

1: √3: 2

Hence proved that the given △ABC is a 30-60-90 Triangle.

Given the length of one side of a triangle, we can find the other side(s) without using long-step methods such as Pythagorean Theorem and trigonometric functions.

Solving a 30-60-90 triangle can have four possibilities:

**Possibility 1**: When the shorter side is known, we can find the longer side by multiplying the shorter side by √3. The hypotenuse can then be obtained by Pythagoras Theorem.

Thus,

Longer side = shorter side × √3

**Possibility 2**: When the longer side is known, we can find the shorter side by dividing the longer side by √3. The hypotenuse can then be obtained by Pythagoras Theorem.

Thus,

Shorter side = longer side/√3

**Possibility 3**: When the shorter side is known, we can find the hypotenuse by multiplying the shorter side by 2. The longer side can then be obtained by Pythagoras Theorem.

Thus,

Hypotenuse = shorter side × 2

**Possibility 4**: When the hypotenuse is known, we can find the shorter side by dividing the hypotenuse by 2. The longer side can then be obtained by Pythagoras Theorem.

Thus,

Shorter side = hypotenuse/2

The formulas of a 30-60-90 triangle when the length of the shorter side is x units are given below:

Let us solve some examples to understand the concepts better.

**A right triangle whose one angle is 60 degrees has the longer side of 12√3 cm. Find the length of its shorter side and the hypotenuse.**

Solution:

Since, the given triangle is a 30-60-90 triangle,

Its side lengths = x: x√3: 2x, here x√3 = longer side = 12√3 cm

Thus,

x√3 =12√3

Squaring both sides we get,

⇒ (x√3)^{2} = (12√3)^{2} cm

⇒ 3x^{2} = 144 x 3

⇒ x^{2} = 144

⇒ x = 12 cm

Hypotenuse = 2x = 2 x 12 = 24 cm Hence the shorter side is 12cm and hypotenuse is 24 cm.

**A right triangle whose one angle is 60 degrees has the longer side of 12√3 cm. Find the length of its shorter side and the hypotenuse.**

Solution:

Since, the given triangle is a 30-60-90 triangle,

Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x√3 = longer side = 12√3 cm

Thus,

x√3 =12√3

Squaring both sides we get,

⇒ (x√3)^{2} = (12√3)^{2} cm

⇒ 3x^{2} = 144 x 3

⇒ x^{2} = 144

⇒ x = 12 cm

Hypotenuse = 2x = 2 x 12 = 24 cm Hence the shorter side is 12cm and hypotenuse is 24 cm.

**The diagonal of a right triangle is 14 cm, find the lengths of the other two sides of the triangle given that one of its angles is 30 degrees.**

Solution:

Since, the given triangle is a 30-60-90 triangle,

Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here 2x = diagonal = hypotenuse = 12√3 cm

Thus,

⇒ 2x = 14 cm

⇒ x = 7 cm

Substituting the value of x, we get,

Longer side = x√3 = 7√3 cm

Hence, the length of the longer side is 7√3 cm**.**

**Find the value of y and z in the given diagram.**

Solution:

Since, the given triangle is a 30-60-90 triangle,

The side measuring 18m is the shorter side because it is opposite the 30-degree angle.

Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x = 18m

Substituting the value of x, we get,

Given,

x√3 = y

Longer side = y = 18√3m

Given,

2x = z

Hypotenuse = z = 2 x 18 = 36m

Hence, the length of the hypotenuse is 36m.

**If one angle of a right triangle is 30° and the measure of the shortest side is 11cm. Find the measure of the remaining two sides.**

Solution:

Since, the given triangle is a 30-60-90 triangle,

Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x = 7cm

Substituting the value of x, we get,

Longer side = x√3 = 7√3cm

Hypotenuse = 2x = 2 x 7 = 14cm

Hence, the length of the other two sides is 7√3cm and 14cm.

**A ramp making an angle of 30 degrees with the ground is used to offload a lorry that is 8 m high. Calculate the length of the ramp.**

Solution:

Since, the ramp makes a 30-60-90 triangle with the ground,

Here,

Shorter side = x = 8m

Since, the length of the ramp is the hypotenuse of the given 30-60-90 triangle,

2x = 2 x 8 = 16m

Hence the length of the ramp is 16m.

Last modified on August 3rd, 2023