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 900
Short leg is the side across 300
SL = ½ x H
SL = ½ x 12 = 6cm
Long leg is the side across 600
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 = 900
BC = 7cm which is the one half of hypotenuse.
So, short leg = SL = 7cm
∴ ∠A = 300
AB= 7√3cm which is √3 times SL
∴ ∠C = 600 Special Right Triangles
