Question

Activities-
(1) What is Pythagoras' relation? And explain Pythagorean triples? (2 Marks)

(2) What is the length of the fourth side of the quadrilateral shown below? (3 Marks)

Answer 1

(1) Pythagoras' relation
The square of the longest side of a right triangle is equal to the sum of the squares of the other two sides

Putting it in reverse, if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then it is a right triangle.

That is, the property of having the square of one side equal to the sum of the squares of the other two, is peculiar to right triangles.
For example, since
$3^2+4^2=5^2$, a triangle with lengths of sides $3,4,5$ are right angle triangles.
What if the Lengths of sides are $6,8,10?$ These sides also make a right angle triangle.
Pythagorean triples
$\begin{array}{cc}& 3^2+4^2=25=5^2\\ & 5^2+{12}^2=169={13}^2\\ & 8^2+{15}^2=289={17}^2\end{array}$
and so on.
Any three natural numbers, with the sum of the squares of two of them equal to the square of the third, is called a Pythagorean triple.
Some examples of Pythagorean triples are
$3,4,5$
$5,12,13$
$8,15,17$

(2) Here quadrilateral is combination of two shapes i.e. is rectangle and right angle triangle
The rectangle length and right angle triangle base are equal. I.e. $24cm$
Height of the right angle triangle is quadrilateral height less than the rectangle width
Thus $15cm-8cm=7cm$
We know that as per Pythagoras' relation, The Square of the longest side of a right triangle is equal to the sum of the squares of the other two sides

Here "c" is the unknown side of the right angle triangle a and b are the known of the known sides the triangle
$$\begin{array}{cc}{c}^2& =a^2+b^2\\ c^2& ={24}^2+7^2\\ & =576+49\\ c& =\sqrt{625}=25\end{array}$$
The diagram shows the unknown side of the right angle triangle is the fourth side of the quadrilateral i.e $25cm$.