Instruction

1

The sides of a right triangle adjacent to the right angle (AB and BC are called the legs. Side lying opposite the right angle is called the hypotenuse (AC).

Let we know the hypotenuse AC of the right triangle ABC: AC| = c. We denote the angle with vertex at point A as a ∟α, the angle with vertex at point B as ∟β. We need to find the lengths |AB| and |BC| of the other two sides.

Let we know the hypotenuse AC of the right triangle ABC: AC| = c. We denote the angle with vertex at point A as a ∟α, the angle with vertex at point B as ∟β. We need to find the lengths |AB| and |BC| of the other two sides.

2

Let known one of the legs of a right triangle. Assume |BC| = b. Then we can use the Pythagorean theorem that the hypotenuse squared is equal to the sum of the squares of the legs: a^2 + b^2 = c^2. From this equation we find unknown side |AB| = a = √ (c^2 - b^2).

3

Let you know one of the angles of a right triangle, suppose ∟α. Then the other two sides AB and BC of the triangle ABC can be found using trigonometric functions. So we get: sine ∟α equal to the ratio of the opposite leg to the hypotenuse sin α = b / c cosine ∟α equal to the ratio of adjacent sides to the hypotenuse cos α = a / c. Hence, we find the required lengths of the sides: |AB| = a = C * cos α, |BC| = b = c * sin α.

4

Let you know the ratio of sides k = a / b. Solve problems using trigonometric functions. The ratio a / b is none other than the cotangent ∟α: the ratio of the adjacent leg to the opposite ctg α = a / b. In this case, from this equality we Express a = b * ctg α. And substitute in the Pythagorean theorem a^2 + b^2 = c^2:

b^2 * ctg^2 α + b^2 = c^2. Make b^2 outside the brackets, we get b^2 * (ctg^2 α + 1) = c^2. And easily get the length of leg b = c / √(ctg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.

By analogy, if you know the ratio of each b / a, we solve the problem with the use of trigonometric functions the tangent tg α = b / a. Substitute the value b = a * tg α in the Pythagorean theorem a^2 * tg^2 α + a^2 = c^2. Hence a = c / √(tg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.

b^2 * ctg^2 α + b^2 = c^2. Make b^2 outside the brackets, we get b^2 * (ctg^2 α + 1) = c^2. And easily get the length of leg b = c / √(ctg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.

By analogy, if you know the ratio of each b / a, we solve the problem with the use of trigonometric functions the tangent tg α = b / a. Substitute the value b = a * tg α in the Pythagorean theorem a^2 * tg^2 α + a^2 = c^2. Hence a = c / √(tg^2 α + 1) = c / √(k^2 + 1), where k is a predetermined ratio of the legs.

5

Consider special cases.

∟α = 30°. Then |AB| = a = c * cos α = c * √3 / 2; |BC| = b = c * sin α = c / 2.

∟α = 45°. Then |AB| = |BC| = a = b = c * √2 / 2.

∟α = 30°. Then |AB| = a = c * cos α = c * √3 / 2; |BC| = b = c * sin α = c / 2.

∟α = 45°. Then |AB| = |BC| = a = b = c * √2 / 2.

Note

Square roots are extracted with a positive sign, because length cannot be negative. It seems obvious, but this error is very common, if to solve the problem automatically.

Useful advice

For finding the legs of a right triangle is convenient to use formula bring: sin β = sin (90° - α) = cos α; cos β = cos (90° - α) = sin α.

# Advice 2: How to find the length of a side of a right triangle

Rectangular is a triangle whose one of the angles is a straight line. Side

**of the triangle**located opposite the right angle is called the hypotenuse and the other two**sides**the legs. To find the lengths of the sides of a rectangular**triangle**, you can use several ways.Instruction

1

You can find out the size of the third

b2 = a2 - c2;

c2 = a2 - b2

For example, a rectangular

S2 = 182 - 142 = 324 - 196 = 128 cm

c = √128 cm

Answer: the length of the second leg is √128 cm or approximately 11.3 cm

**parties**, knowing the lengths of two other sides**of the triangle**. This can be done by using the Pythagorean theorem, which States that the square of the hypotenuse of a rectangular**triangle**is equal to the sum of the squares of the other two sides. (a2 = b2+ c2). Hence, we can Express the lengths of all sides of a rectangular**triangle**:b2 = a2 - c2;

c2 = a2 - b2

For example, a rectangular

**triangle**are known, the length of the hypotenuse a (18 cm) and one of the legs, for example c (14 cm). To find*the length of the*other leg, you want to make 2 algebraic operations:S2 = 182 - 142 = 324 - 196 = 128 cm

c = √128 cm

Answer: the length of the second leg is √128 cm or approximately 11.3 cm

2

You can resort to another method if you know the length of the hypotenuse and the magnitude of one of the acute angles of the rectangular

a = s*sinα;

b = s*cosα.

Example: the length of the hypotenuse is 15 cm, one of the acute angles is 30 degrees. To find the lengths of the other two sides you need to perform 2 steps:

a = 15*sin30 = 15*0.5 = 7.5 cm

b = 15*cos30 = (15*√3)/2 = 13 cm (approx)

**triangle**. Let the length of the hypotenuse equal to c, one of the acute angles equal to α. In this case, find the 2 other**side of**rectangular**triangle**will be using the following formulas:a = s*sinα;

b = s*cosα.

Example: the length of the hypotenuse is 15 cm, one of the acute angles is 30 degrees. To find the lengths of the other two sides you need to perform 2 steps:

a = 15*sin30 = 15*0.5 = 7.5 cm

b = 15*cos30 = (15*√3)/2 = 13 cm (approx)

3

The most trivial way to find

P = a + b + c, where P is the perimeter of a rectangular

*the length of***sides**of a rectangular**triangle**is to Express it from the perimeter of this shape:P = a + b + c, where P is the perimeter of a rectangular

**triangle**. From this expression it is easy to Express*the length of*any of sides of a rectangular**triangle**.# Advice 3: What are the sides of a right triangle

The amazing properties of right triangles people interested in antiquity. Many of these properties were described by Greek scientist Pythagoras. In Ancient Greece appeared, and the names of the sides of a right triangle.

## What kind of triangle is called rectangular?

There are several types of triangles. Some have all the angles acute, the other one obtuse and two acute, the third – two sharp and straight. On this basis, each type of these geometric figures and received a name: acute-angled, obtuse-angled and rectangular. That is, rectangle is a triangle whose one angle is 90°. There is another definition, similar to the first. Is called a rectangular triangle whose two sides are perpendicular.

## The hypotenuse and the legs

At acute and obtuse triangles to segments connecting the vertices of the angles, referred to as " the parties. From the rectangular triangle the parties have other names. Those that are adjacent to the right angle are called the legs. Side opposite the right angle is called the hypotenuse. In Greek the word "hypotenuse" means "stretched", and "side" - perpendicular.

## The ratio between the hypotenuse and the legs of

The sides of a right triangle are linked by defined relationships, which greatly facilitates calculations. For example, knowing the sizes of the other two sides, you can calculate the length of the hypotenuse. This ratio is named opened his math called Pythagorean theorem, and it looks like this:

c2=a2+b2, where C is the hypotenuse, a and b are the legs. That is, the hypotenuse will equal the square root of the sum of the squares of the other two sides. To find any of the sides, enough of the square of the hypotenuse subtract the square of the other leg and removing from the obtained difference square root.

## The adjacent and opposite side

Draw a right triangle ACB. Letter C to denote a vertex of a right angle, and the vertices of the acute angles. The side opposite each corner, conveniently called a, b and C, by the names in front of them reclining angles. Consider the angle A. the Leg and for him to be the opposite, side b – yard. The ratio of the opposite leg to the hypotenuse is called the sine. To calculate the trigonometric function by the formula: sinA=a/c. The ratio of adjacent sides to the hypotenuse is called the cosine. It is calculated by the formula: cosA=b/c.

Thus, knowing the angle and one side, you can use the following formulas to calculate the other side. Trigonometric ratios are connected both sides. The ratio of opposite over adjacent to is called the tangent, and adjacent to the opposite – cotangent. To Express these relationships can be a formula tgA=a/b or ctgA=b/a.