Question

Pythagorean Theorem
adjacent arcs
center
central angle
circle
circumference
concentric circles
congruent arcs
congruent circles
diameter
major arc
minor arc
pi
radius
semicircle
use one of these to prove what you learned in this unit

Answer 1

Using the Pythagorean Theorem, we can prove that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's consider a right-angled triangle with sides of length a, b, and c (where c is the hypotenuse). According to the Pythagorean Theorem, we have:

a² + b² = c²

Now, let's draw a circle with radius r centered at point O. Let's draw a right-angled triangle within the circle such that one of its sides is the radius (r), and the other two sides are a secant and a tangent to the circle as shown below:

[diagram of a circle with a right-angled triangle inscribed within it]

According to the properties of circles, the angle between a radius and a tangent line at the point of contact is always a right angle. Therefore, the triangle we have created inside the circle is a right-angled triangle.

Now, let the lengths of the two sides of the triangle be a and b, and the length of the radius be c. Applying the Pythagorean Theorem to this triangle, we have:

a² + b² = c²

This proves that the Pythagorean Theorem holds true not only for traditional right-angled triangles but also for triangles inscribed in circles.