Therefore, the length of the diagonal is 15 cm. We have:Īdding and taking the square root, we get: Then, we can use the distance formula to find the distance between $A$ and $C$, which is equal to the diagonal. We can assign coordinates to the vertices of the rectangle, such as $A(0, 0)$, $B(12, 0)$, $C(12, 9)$, and $D(0, 9)$.
Therefore, the length of the diagonal is 15 cm.Īlternatively, we can use the distance formula to find the same answer. Taking the square root of both sides, we get: Using the Pythagorean theorem, we can write: The diagonal is the hypotenuse of this triangle, and its length is denoted by $d$. We can draw a right triangle inside the rectangle, as shown below: Suppose we have a rectangle with sides of length 12 cm and 9 cm.
FINDING LENGTH OF DIAGONAL OF A RECTANGLE HOW TO
Here are some examples of how to use the Pythagorean theorem and the distance formula in different situations.Įxample 1: Finding the Length of a Diagonal of a Rectangle For example, we can use them to find the length of a diagonal of a rectangle, the height of a building, the speed of a moving object, the angle of elevation or depression, the area of a circle, and many more. The Pythagorean theorem and the distance formula are useful tools for solving many problems in mathematics, physics, engineering, and other fields. The distance formula can be seen as a special case of the Pythagorean theorem when we consider a right triangle whose vertices are given by coordinates.
The Pythagorean theorem applies to any right triangle, while the distance formula applies to any pair of points in a plane. The Pythagorean theorem and the distance formula are related because they both use squares and square roots to express geometric relationships. It shows that we can calculate the distance between any two points in a plane by using their coordinates and applying some basic arithmetic operations. Subtracting $2ab$ from both sides, we obtain: Therefore, we have:Įxpanding and simplifying this equation, we get: The area of each square is equal to the square of its side length. One of the simplest proofs is based on rearranging four copies of a right triangle into two different squares, as shown below: The Pythagorean theorem can be proven in many ways, using geometry, algebra, or calculus. Where $a$ and $b$ are the lengths of the legs (the sides adjacent to the right angle) and $c$ is the length of the hypotenuse. In any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. According to legend, he discovered this theorem by observing the patterns of tiles on the floor of a temple. The Pythagorean theorem is named after the Greek philosopher and mathematician Pythagoras, who lived around the 6th century BC.
FINDING LENGTH OF DIAGONAL OF A RECTANGLE FREE
The internal diagonal is \( 5\sqrt\)Ī is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning.
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