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# Right Triangle Properties

### The pythagorean theorem

### Converse of the pythagorean theorem

### Median throrem of a right triangle

### Property of the 30^{o}-60^{o}-90^{o} right triangle

### Corollary of the 30^{o}-60^{o}-90^{o} right triangle

### The 45^{o}-45^{o}-90^{o} right triangle

### Definition of sine, cosine and tangent<

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.

In right triangle ABC, AC^{2} = AB^{2} + BC^{2}. 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. hypotenuse^{2} = Leg_{1}^{2} + Leg_{2}^{2}

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.

In the figure above, if AB^{2} = AC^{2} + BC^{2}, then the triangle ABC is a right triangle. or if c^{2} = a^{2} + b^{2}, 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.

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.

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.

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.

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

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

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.

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

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

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.

- In right triangle ABC, since angle A = 45
^{o} - So angle B = 90
^{o}- 45^{o}= 45^{o} - Since angle A = angle B = 45
^{o} - So AC = BC (In a triangle, equal angles opposite equal sides.)
- In right triangle ABC, AB is the hypotenuse. AC and BC are legs.
- AB
^{2}= AC^{2}+ BC^{2} - Since AC = BC, so
- AB
^{2}= 2AC^{2} - 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.

**Example 1**

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

- Solution
- In the right triangle ABC, since angle B = 30
^{o}and angle B opposite AC, - and since angle C = 90
^{o}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 30
^{o}is one-half the length of the hypotenuse.) - AB
^{2}= AC^{2}+ BC^{2}(the pythagorean theorem) - BC
^{2}= AB^{2}- AC^{2}= 6^{2}- 3^{2}= 36 - 9 = 27 = 3^{2}× 3 - BC = 3(square root of 3)

**Example 2**

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

- Solution
- In right triangle ABC, since angle B = 45
^{o}and angle C = 90^{o}, - so angle A = 90
^{o}- 45^{o}= 45^{o} - so AC = BC = 5 (In a triangle, equal angles opposite equal sides)
- then AB
^{2}= AC^{2}+ BC^{2}= 2AC^{2}= 2 × 5^{2} - so AB = 5(square root of 2)

**Example 3**

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

- 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 = 30
^{o}and AD = 2, - so AC = 2 AD = 2 × 2 = 4
- CD
^{2}= AC^{2}- AD^{2}= 4^{2}- 2^{2}= 16 - 4 = 12 = 3 × 2^{2} - CD = 2(square root of 3)
- In right triangle ABD,
- since angle B = 45
^{o}, so angle BAD = 45^{o} - 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)