How Long Does the Hypotenuse of a Right Triangle Measure
— Find it Using Area-based Approach!
We will have an exciting journey of discovery, following the footstep of early pioneers in math.
Let the problem be raised as:
Problem Given two legs of a right triangle, what is the length of its hypotenuse?
After a bit thought, we decide to reduce the problem to the following form:
Standardized Problem Given a right triangle with one leg being 1, the other leg being x, what
length of the hypotenuse?
With familiarity with concepts of similar triangles and proportion, you will find that solution to
standardized problem leads immediately to a solution for the original one. Even without that,
the connection of the two problems can be intuitively understood. Suppose a triangle with two legs:
1 and x, and a hypotenuse of y, then we know a triangle with two legs 2, 2x will have a hypotenuse of 2y.
For this reason, below we will focus only on the standardized problem. The goal is to fill out a form
where leg one is always 1, leg two is any integer numbers: 1, 2, 3, .. etc.
This provides us with the length of the hypotenuse.
[table]
Leg-1(a),Leg-2(b),Hypotenuse(c)
1,1,?
1,2,?
1,3,?
1,4,?
[/table]
Let’s set out to work!
For row 1 – Leg two equals Leg one Equals 1
This is the case for a right isosceles triangle.
The right isosceles triangle is shown. Reflect it twice, to the horizontal leg and respectively
to vertical leg, as line symmetry. Then rotate the original triangle
around the right-angle corner for 180 degrees. In such way we obtain three new
triangles. The new three and the original one together form one square.
Since the original triangle has an area of (1/2) × 1 × 1 = (1/2), the four triangles have a total area of
4 × (1/2) = 2. While they together form one square with a side to be decided; let us suppose it to be y.
Then the area of square must equal 2. So y2 = 2.
We will simply write for this case as Leg one = 1, Leg two = 1, and the Hypotenuse is sqrt 2.