You will need

- - Knowledge of two of the three sides of a right triangle;
- - Knowledge of the angles of a triangle.

Instruction

1

Method 1. The use of the Pythagorean theorem. The theorem States that the hypotenuse squared is equal to the sum of the squares of the legs. It follows that any of the sides of a right triangle can be calculated, knowing the two other sides (Fig.2)

2

Method 2. Follows from the fact that the median drawn from the right angle to the hypotenuse forms between the 3 similar triangles (Fig. 3). In this figure, are similar triangles ABC, BCD and ACD.

Note

When calculating sides of a right triangle can play knowing the signs:

1) If the leg of a right angle is opposite the angle of 30 degrees, it is equal to half the hypotenuse;

2) the Hypotenuse is always longer than any of the other two sides;

3) If you are around the right triangle described a circle, its centre must lie in the middle of the hypotenuse.

1) If the leg of a right angle is opposite the angle of 30 degrees, it is equal to half the hypotenuse;

2) the Hypotenuse is always longer than any of the other two sides;

3) If you are around the right triangle described a circle, its centre must lie in the middle of the hypotenuse.

# Advice 2 : How to find the hypotenuse on the side and corners

Is called the hypotenuse side in a right triangle that is opposite the angle of 90 degrees. In order to calculate its length, enough to know the length of one of the legs and the size of one of the acute angles of the triangle.

Instruction

1

At a known side and an acute angle of a right triangle, the hypotenuse can be equal to the ratio of the leg to the cosine/sine of that angle if the given angle is opposite him/yard:

h = C1(or C2)/sinα;

h = C1(or C2)/cosα.

Example: suppose that we are given right triangle ABC with hypotenuse AB and a right angle C. Let angle B equal 60 degrees, and angle a of 30 degrees Length of side BC is 8 cm Need to find the length of the hypotenuse AB. For this you can use any of the above methods:

AB = BC/cos60 = 8 cm.

AB = BC/sin30 = 8 cm.

h = C1(or C2)/sinα;

h = C1(or C2)/cosα.

Example: suppose that we are given right triangle ABC with hypotenuse AB and a right angle C. Let angle B equal 60 degrees, and angle a of 30 degrees Length of side BC is 8 cm Need to find the length of the hypotenuse AB. For this you can use any of the above methods:

AB = BC/cos60 = 8 cm.

AB = BC/sin30 = 8 cm.

# Advice 3 : How to calculate the hypotenuse

Call the hypotenuse side of a right triangle that lies opposite the right angle. It is the largest side of a right triangle. To be calculated by the Pythagorean theorem, or by using the formulas of trigonometric functions.

Instruction

1

The other two sides are called the sides of a right triangle adjacent to the right angle. In the figure, the legs are designated as AB and BC. Imagine you are given the lengths of both legs. Denote them as |AB| and |BC|. In order to find the length of the hypotenuse |AC|, we will use the Pythagorean theorem. According to this theorem the sum of the squares of the legs equals the square of the hypotenuse, i.e., in the notation of our pattern |AB|^2 + |BC|^2 = |AC|^2. From the formula we find that the length of the hypotenuse AC is as |AC| = √(|AB|^2 + |BC|^2) .

2

Let's consider an example. Imagine you are given the lengths of the legs |AB| = 13, |BC| = 21. By the Pythagorean theorem we see that |AC|^2 = 13^2 + 21^2 = 169 + 441 = 610. In order to obtain the length of the hypotenuse, you must extract the square root of the sum of the squares of the other two sides, i.e., number 610: |AC| = √610. Using the table of squares of integers, it appears that the number 610 is not a complete square of any integer. In order to obtain the final value of the length of the hypotenuse, try to make a full square under the root sign. To do this, divide the number 610 to multipliers. 610 = 2 * 5 * 61. The table of Prime numbers looking 61 is a Prime number. Therefore, further reduction of the number √610 impossible. Get the final answer |AC| = √610.

If the square of the hypotenuse was equal to, for example, is 675, then √675 = √(3 * 25 * 9) = 5 * 3 * √3 = 15 * √3. In the event that such a cast may perform a reverse check - raise the result in the square and compare it to the original value.

If the square of the hypotenuse was equal to, for example, is 675, then √675 = √(3 * 25 * 9) = 5 * 3 * √3 = 15 * √3. In the event that such a cast may perform a reverse check - raise the result in the square and compare it to the original value.

3

Let us known one of the legs and adjacent to it a corner. For definiteness, let it be a leg |AB and angle α. Then we can use the formula for the trigonometric functions cosine the cosine of an angle is equal to the ratio of the adjacent leg to the hypotenuse. I.e., in our notation cos α = |AB| / |AC|. Hence we obtain the length of the hypotenuse |AC| = |AB| / cos α.

If we know the side |BC|, and the angle α, we use the formula for calculating the sine of an angle is the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| / cos α.

If we know the side |BC|, and the angle α, we use the formula for calculating the sine of an angle is the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| / cos α.

4

For clarity, consider an example. Given the length of the leg |AB| = 15. And the angle α = 60°. Get |AC| = 15 / cos 60° = 15 / 0.5 = 30.

Consider how you can check your result by using the Pythagorean theorem. To do this we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tg α = |BC| / |AC|, we get |BC| = |AB| * tg α = 15 * tg of 60° = 15 * √3. Next apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. The validation is completed.

Consider how you can check your result by using the Pythagorean theorem. To do this we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tg α = |BC| / |AC|, we get |BC| = |AB| * tg α = 15 * tg of 60° = 15 * √3. Next apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. The validation is completed.

Useful advice

Calculating the hypotenuse, do the test - whether the value obtained using the Pythagorean theorem.