Multiple choice

An isosceles $\Delta ABC$ is inscribed in a circle. If AB= AC =$12 \sqrt{5}$ cm and BC = 24 cm. The radius of the circle is

  1. $30 cm$
  2. $20cm$
  3. $15cm$
  4. $25cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In an isosceles triangle with sides 12*sqrt(5), 12*sqrt(5), and base 24, the height h to the base is sqrt((12*sqrt(5))^2 - 12^2) = sqrt(720 - 144) = sqrt(576) = 24. The circumradius R = (abc) / (4 * Area). Area = 1/2 * base * height = 1/2 * 24 * 24 = 288. R = (12*sqrt(5) * 12*sqrt(5) * 24) / (4 * 288) = (144 * 5 * 24) / 1152 = 17280 / 1152 = 15.

AI explanation

For a triangle with sides a, b, and c, the circumradius R is found using the formula R = (a * b * c) / (4 * Area). Dropping a height to the base BC of 24 cm creates a right triangle with hypotenuse 12 * sqrt(5), giving a height of sqrt((12 * sqrt(5))^2 - 12^2) = 24 cm and an area of (1/2) * 24 * 24 = 288 sq cm. Plugging the side lengths into the circumradius formula gives (24 * 12 * sqrt(5) * 12 * sqrt(5)) / (4 * 288) = 15 cm.

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