a 30°- 60°- 90° triangle, the side opposite to 30° is 5. Find the length of side opposite to 60° and the hypotenuse respectively.
Answers
Step-by-step explanation:
Because of the geometry of the triangle. This can be seen using the Pythagorean Therom for right triangles: a^2+b^2=c^2 where c is the length of the side opposite the 90 degree angle and a and b are the lengths of the other two sides.
Putting two 30–60–90 triangles back to back forms a triangle with 3 60 degree angles, which makes it an equilateral triangle with side length 2x (assuming x is the length of the side opposite the 30 degree angle):

The vertical line down the middle of this triangle bisects the top angle making two 30 degree angles, and is perpendicular to the bottom edge of the triangle making the 90 degree angles for the two 30–60–90 triangles shown. Using Pythagorean’s therom to solve for the length of this middle line (and the length of the side opposite the 60 degree angle in the 30–60–90 triangle) gives:
if a is the side opposite the 30 degree angle and b is opposite the 60 degree angle
a=x as shown
c=2x as shown
a^2+b^2=c^2
Solving for b gives:
b=sqrt(c^2-a^2)=sqrt((2x)^2-(x)^2)=sqrt(4x^2-x^2)=sqrt(3x^2)=(x)(sqrt(3))
which is the length of the side opposite the 60 degree angle in a 30–60–90 triangle.
Given: In a 30°- 60°- 90° triangle, the side opposite to 30° is 5.
To find: The length of side opposite to 60° and the hypotenuse
Solution: Let us imagine a triangle ABC where A = 60°, B = 90° and C = 30°.
Hence, AB = 5 as it is opposite to C (=30°).
Now let's find the length of the side opposite to 60°(angle A), that is, side BC.
∴ tanA = BC/AB
⇒ tan 60° = BC/ 5
⇒ √3 = BC/5
⇒ BC = 5√3
Now we need to find the hypotenuse, that is, side AC.
AC² = AB² + BC²
⇒ AC² = 5² + (5√3)²
⇒ AC² = 25 + 75
⇒ AC² = 100
⇒ AC = 10
Answer: length of the side opposite to 60° = 5√3
length of the hypotenuse = 10