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GENERATE WORK

GENERATE WORK

**Input Data :**

`a` = 3

`b` = 4

**Objective :**

Find the unknown length `c` of right triangle.

**Formula :**

`c = \sqrt(a^2 + b^2)`

**Solution :**

`c = \sqrt(3^2 + 4^2)`

` = \sqrt(9 + 16)^`

` = \sqrt(25)`

`c = 5`

** Pythagorean Theorem calculator** calculates the length of the third side of a right triangle based on the lengths of the other two sides using the Pythagorean theorem. In other words, it determines:

- The length of the hypotenuse of a right triangle, if the lengths of the two legs are given;
- The length of the unknown leg, if the lengths of the leg and hypotenuse are given.

It is necessary to follow the next steps:

- Enter the lengths of two sides of a right triangle in the box. These values must be positive real numbers or parameters. Note that the length of a segment is always positive;
- Press the "
**GENERATE WORK**" button to make the computation; - Pythagorean theorem calculator will give the length of the third side of a right triangle.

In a right triangle $\Delta ABC$, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, i.e.
$$c^2=a^2+b^2,$$
where $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the legs of $\Delta ABC$.

We must make sure that the length of the hypotenuse is labeled by $c$ and the lengths of the legs are labeled by $a$ and $b$.

If we know the length of the hypotenuse and the length of one leg, then the length of the other leg can be calculated by the formula $$a^2=c^2-b^2$$ or $$b^2=c^2-a^2$$

We must make sure that the length of the hypotenuse is labeled by $c$ and the lengths of the legs are labeled by $a$ and $b$.

If we know the length of the hypotenuse and the length of one leg, then the length of the other leg can be calculated by the formula $$a^2=c^2-b^2$$ or $$b^2=c^2-a^2$$

One of the most famous and most useful theorems in mathematics is the Pythagoras Theorem. The theorem is named after the Greek mathematician Pythagoras because he gave its first proof, although no evidence of it exists.

**Pythagorean Theorem:** If $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the legs in a right triangle,
then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, i.e.
$$c^2=a^2+b^2$$

Thus, if we know the lengths of two out of three sides in a right triangle, we can find the length of the third side. The Pythagorean Theorem can also be expressed in terms of area. In a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares
on the two legs.

The converse of the Pythagoras Theorem is also valid. The Pythagorean converse theorem can help us in classifying triangles.

- If the square of the length of the longest side of a triangle is equal to the sum of squares of the lengths of the other two sides, then the triangle is a right triangle.
- If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.
- If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle.

In a right triangle $\Delta ABC$, the length of the hypotenuse $c$ is equal to the square root of the sum of the squares of the lengths of the legs $a$ and $b$. That means,

$$c=\sqrt{a^2+b^2}$$

In a right triangle $\Delta ABC$, the length of either leg is equal to the square root of the difference of the squares of the length of the hypotenuse and the length of the other leg.
That means,
$$a=\sqrt{c^2-b^2}\quad{\rm or}\quad b=\sqrt{c^2-a^2}$$

Pythagorean Theorem calculator work with steps shows the complete step-by-step calculation for finding the length of the hypothenuse $c$ in a right triangle $\Delta ABC$ having
the lengths of two legs $a=3$ and $b=4$. For any other combinations of side lengths, just supply lengths of two sides and click on the "GENERATE WORK" button.
The grade school students may use this Pythagoras theorem calculator to generate the work, verify the results or do their homework problems efficiently.
The Pythagoras Theorem is widely applied in mathematics. The formula for finding distance between two points is based on the Pythagorean Theorem.
For any two points $A(x_A,y_A)$ and $B(x_B,y_B)$ in the two-dimensional Cartesian coordinate plane, the formula for distance between these points is derived from the Pythagorean Theorem, i.e.

$${AB}^2=(x_B-x_A)^2+(y_B-y_A)^2$$

Similarly, it is cited in the formula for distance between two points in the three-dimensional Cartesian coordinate space. The Pythagorean Theorem is also useful in finding the area of some polygon.In the theory of numbers, a triple $(a, b, c)$ consisted of three positive integers $a, b,$ and $c,$ which satisfies the Pythagorean formula

$$a^2 + b^2 = c^2$$

is the Pythagorean Triple. If $(a, b, c)$ is a Pythagorean Triple, then so is $(ka, kb, kc$) for any positive integer $k$. For example, the Pythagorean Triples are: $(3, 4, 5),$ $(5, 12, 13),$ $(8, 15, 17),$ $(7, 24, 25),$ $(20, 21, 29),$ etc.In the theory of complex numbers, the modulus $|z|$ of the complex number $z=a+ib$ is determined by

$$|z|=\sqrt{a^2+b^2}$$

These three magnitudes are related by the Pythagorean formula,$$z^2=a^2+b^2$$

It means, the distance of the point $z$ from the origin $O(0,0)$ is $|z|$ in the complex plane.The distance between two complex numbers $z_1=a+ib$ and $z_2=c+id$ in the complex
plane is also connected with the Pythagoras Theorem. Therefore, the distance between two complex numbers is$$|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$$

In the trigonometry, for any acute angle $\angle A\; (m\angle A=\alpha)$ of a right triangle $\Delta ABC$, the Pythagorean Identify is valid
$$\sin^2\alpha+\cos^2\alpha=1$$

**Practice Problem 1:**

A sail on a sailboat is in the shape of a right triangle. The length of the longest side of the sail is $220$ centimeters, and the length of the other side of the sail is $5$ meters.

**Practice Problem 2:**

Given a triangle $\Delta ABC$, as it is shown in the picture below. The altitude $\overline{AA'}$ divides the sides $\overline{BC}$ into
two segments ${BA'}=5$ and ${CA'}=9$. Find the perimeter of the triangle $\Delta ABC$.