# 30-60-90 Triangles

**30-60-90 triangles**

Another type of special right triangles is the 30°- 60°- 90° triangle. This is right triangle whose angles are 30°, 60°and 90°. The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio 1:√3:2

**Side1: Side2 : Hypotenuse = a : a√3 : 2a**

**Some Solved Examples**

**Example 1**: Find the length of the hypotenuse, if the two sides are 5√3 and 5.

**Solution**:

As the two sides fit in the ratio a : a√3 : 2a

a = 5

Hypotenuse = 2a

Hypotenuse = 2 x 5 = 10

**Example 2**: Find x and y

**Solution**:

As the given triangle is a special triangle of 30 - 60 -90.

The ratio for this triangle is Side1: Side2 : Hypotenuse = a : a√3 : 2a

Here hypotenuse = 2a = 16

So, a = 8

y = 8

x = 8√3

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2) In a right triangle, the longest side is 12cm. Find the long leg and short leg.

**Solution :**

Longest side= Hypotenuse (H)= 12 cm (side opposite to 90

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Short leg is the side across 30

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SL = ½ x H

SL = ½ x 12 = 6cm

Long leg is the side across 60

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LL = SL x √3

LL = 6√3 cm

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3)In Δ ABC, AC= 14 cm; BC= 7 cm and AB= 7√3 cm. Find the measures of angles A, B and C.

**Solution**

AC = 14cm; BC = 7cm and AB =7√3cm

From the above information we can see that, AC = 2 x BC and it is longest side than the other two.

So, AC will be the hypotenuse.

The angle between AC is B

So,∠B = 90

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BC = 7cm which is the one half of hypotenuse.

So, short leg = SL = 7cm

∴ ∠A = 30

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AB= 7√3cm which is √3 times SL

∴ ∠C = 60

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**Special Right Triangles**

Special Right Triangles

30-60-90 Triangles

Pythagorean Theorem

Special Right Triangles

30-60-90 Triangles

Pythagorean Theorem