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Right Triangle Properties

The pythagorean theorem

The square of the length of the hypotenuse of a right triangle is equal to the sum of the square of the lengths of the legs.

What is the pythagorean theorem? How to use the pythagorean theorem?

In right triangle ABC, AC2 = AB2 + BC2. In right triangle ABC, there is one hypotenuse which opposite the right angle. There are two legs which are two sides of the right angle. hypotenuse2 = Leg12 + Leg22

Converse of the pythagorean theorem

In a triangle, if the square of the length of a side is equal to the sum of the square of the lengths of other two sides, then the triangle is a right triangle.

How to tell that a triangle is a right triangle?

In the figure above, if AB2 = AC2 + BC2, then the triangle ABC is a right triangle. or if c2 = a2 + b2, then the triangle ABC is a right triangle. Note: in triangle ABC, a is the side BC which opposite the angle A, b is the side AC which opposite the angle B and c is the side AB which opposite the angle C.

Median throrem of a right triangle

In a right triangle, the length of the median of hypotenuse drawn from the hypotenuse to its opposite angle is one-half the length of the hypotenuse.

What is the median throrem? How to use the median throrem?

In the figure above, if triangle ABC is a right triangle, AB is the hypotenuse and D is the midpoint of AB, then CD = (1/2)AB. CD is the median of the hypotenuse.

Property of the 30o-60o-90o right triangle

In a right triangle, if an angle is 30 degrees, then the length of its opposite side is one-half the length of the hypotenuse.

What is the length of the side opposite to 30 degree in a right triangle?

In right triangle ABC, if angle B = 30o, then AC = (1/2)AB.

Note: AB is the hypotenuse of the right triangle ABC. AC is a leg which opposite the angle B.

Corollary of the 30o-60o-90o right triangle

In a right triangle, if the length of a leg is one-half the length of the hypotenuse, then the angle opposite this leg is 30 degrees.

How to tell that an angle is 30 degree in a right triangle?

In right triangle ABC, if AC = (1/2)AB, then the angle B = 30o.

Note: AB is the hypotenuse and AC is a leg. The hypotenuse is opposite the right angle.

The 45o-45o-90o right triangle

In a right triangle, if an angle is 45 degrees, then the length of the hypotenuse is the product of square root 2 and a leg.

What is the length of the leg when an angle is 45 degree in a right triangle?
In right triangle ABC, since angle A = 45o
So angle B = 90o - 45o = 45o
Since angle A = angle B = 45o
So AC = BC (In a triangle, equal angles opposite equal sides.)
In right triangle ABC, AB is the hypotenuse. AC and BC are legs.
AB2 = AC2 + BC2
Since AC = BC, so
AB2 = 2AC2
In a right triangle, if an angle is 45 degrees, then the length of the hypotenuse is the product of square root 2 and a leg.
The length of the hypotenuse is equal to the length of either leg multiplied by square root of 2.
The length of either leg is equal to one-half the length of the hypotenuse multiplied by square root of 2.

Definition of sine, cosine and tangent<

The figure of a right triangle for the definition of sin, cos and tan

What is the definition of the sin A?

What is the definition of the cos A?

What is the definition of tan A

What is the relationship of tan a, sin a and cos a?

Example 1

In right triangle ABC, if the angle B = 30o and AB = 6, what is the length of BC?

example 1 of the right triangle.
Solution
In the right triangle ABC, since angle B = 30o and angle B opposite AC,
and since angle C = 90o and angle C opposite AB, so AB is the hypotenuse,
so AC = (1/2)AB = (1/2) × 6 = 3
(In a right triangle, the length of the leg opposite 30o is one-half the length of the hypotenuse.)
AB2 = AC2 + BC2 (the pythagorean theorem)
BC2 = AB2 - AC2 = 62 - 32 = 36 - 9 = 27 = 32 × 3
BC = 3(square root of 3)

Example 2

In the right triangle ABC, if the angle B = 45o and BC = 5. Find the length of AB.

example 2 of the right triangle.
Solution
In right triangle ABC, since angle B = 45o and angle C = 90o,
so angle A = 90o - 45o = 45o
so AC = BC = 5 (In a triangle, equal angles opposite equal sides)
then AB2 = AC2 + BC2 = 2AC2 = 2 × 52
so AB = 5(square root of 2)

Example 3

In the triangle ABC, angle B = 45o, angle C = 30o and AD is perpendicular to BC at D. AD = 2, find the length of BC. (point D lies on BC)

How to find the lengths of the legs in a right triangle?
Solution
Since AD is perpendicular to BC at point D (Given), so
triangle ABD and triangle ACD are right triangles.
In right triangle ACD,
since angle C = 30o and AD = 2,
so AC = 2 AD = 2 × 2 = 4
CD2 = AC2 - AD2 = 42 - 22 = 16 - 4 = 12 = 3 × 22
CD = 2(square root of 3)
In right triangle ABD,
since angle B = 45o, so angle BAD = 45o
so BD = AD = 2 (triangle ABD is an isosceles right triangle)
BC = BD + CD = 2 + 2(square root of 3) = 2(1 + square root of 3)