The distance from the center of a circle to a chord lengh of 16 cm is 15 cm. About the circle is circumscribed a triangle with perimeter of 2 m. Calculate the area of the triangle.
(1) We are given that the distance from the center of the circle to a chord of length 16cm is 15cm.
Now a diamter drawn perpendicular to a chord bisects the chord. So draw a diamter perpendicular to the chord. This creates a right triangle; one leg is 15cm (given), one leg is 8cm (chord is bisected), and the hypotenuse is the radius of the circle. We can regonize that 8-15-17 is a primitive Pythagorean triple, or use the Pythagorean theorem `r^2=15^2+8^2=289 ==> r=17` .
The radius of the circle is 17cm.
(2) If the triangle is equilateral, then the area can be found by `A=1/2ap` where a is the apothem and p is the perimeter. The perimeter is given as 2m=200cm. The apothem will be the radius of the circle. (The apothem is the segment drawn from the center of a regular polygon perpendicular to a side. The center of the triangle will be the center of the inscribed circle, and since a radius drawn to a point of tangency is perpendicular to the tangent this is the apothem.)
So the area of the triangle will be `A=1/2(200)(17)=1700"cm"^2`