The Pythagorean relationship is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that 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. This relationship is expressed by the equation \(a^2 + b^2 = c^2\text{,}\) where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two sides.
The longest side of the right triangle (the side opposite the \(90^\circ\) angle) is called the hypotenuse and the other two (shorter) sides are called the legs of the triangle.
We use the Pythagorean relationship in many real-life situations, such as:
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{\color{green} \text{Carpentry}}\text{: To design classrooms, furniture and divide land into plots.}
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{\color{green}\text{ Building and Room Design}}\text{: For example, when building classrooms or arranging furniture.}
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{\color{green} \text{Sports}}\text{: In games like basketball, soccer, or football, it helps measure distances, like how far a player runs or the shortest path to the goal.}
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\({\color{green} \text{If you make squares on each side of a right-angled triangle,
the area of the two smaller squares adds up to the area of the largest square.}}\)
Cherotich and Onyango are Grade 7 learners. During the August holiday, they visited their grandmother, who lives nearby a village. Their grandmother is a grocer. Cherotich and Onyango help there grandmother to fetch fruits from a tree for selling. Cherotich used a \(10\,m\) ladder to climb the fruit tree. The ladder had its foot \(6\,m\) from the bottom of the tree and leaned on the tree at the height of \(8\,m\text{.}\)
From the figure alongside, side \(a\) is the \({\color{blue}\text{base}}\text{,}\) side b is the \({\color{blue}\text{height}}\) and side c is the \({\color{blue}\text{hypotenuse}}\text{.}\)Therefore the relationship is, \(a^2+b^2=c^2\text{.}\)
The base of a ladder is \(5\,m\) from the base of a vertical wall. A painter places the ladder such that it touches the top of the wall at a point \(12\,m\) above the ground.calculate the length of the ladder (c).
