Isosceles Triangle ABC is inscribed in a circle with radius 12cm. Aide AB is a diameter. Determine the perimeter of the triangle, correct to one decimal place.
Point C must be midway between A and B on the circle since the triangle is isosceles. ABC therefore has angles of 45, 90 and 45 degrees. AB has length 12 cm. AC and BC have lengths
6/sin45 = 6*2/sqrt2 = 6 sqrt2
The total perimeter is 12 + 12sqrt2 =
12(1 + sqrt2)= 29.0 cm
Drawing the figure yourself should help explain where I got the above numbers.
To find the perimeter of the isosceles triangle ABC, we need to know the lengths of its sides.
Since AB is a diameter of the circle, it passes through the center of the circle. Therefore, triangle ABC is a right triangle, with the right angle at vertex C.
The triangle ABC is isosceles, which means that two of its sides are equal in length. Since AB is the diameter of the circle, it is twice the length of the radius, which is equivalent to 2 * 12 cm = 24 cm.
Let's denote the length of the equal sides as x. By using the Pythagorean Theorem, we can find x:
AB^2 = AC^2 + BC^2
(24 cm)^2 = x^2 + x^2
576 cm^2 = 2x^2
x^2 = 576 cm^2 / 2
x^2 = 288 cm^2
x ≈ √288 ≈ 16.9706 cm (rounded to four decimal places)
Now, we know that the length of the equal sides of triangle ABC is approximately 16.9706 cm. To find the perimeter, we add up the lengths of all three sides:
Perimeter = AB + AC + BC
Perimeter = 24 cm + 16.9706 cm + 16.9706 cm
Perimeter ≈ 57.9412 cm (rounded to one decimal place)
Therefore, the perimeter of the triangle ABC is approximately 57.9 cm.