A circle is divided into four sectors such that the proportions of their areas are 1 : 2 : 3 : 4. What is the angle of the largest sector ?<\/p>\n
[amp_mcq option1=”120\u00b0” option2=”125\u00b0” option3=”136\u00b0” option4=”144\u00b0” correct=”option4″]<\/p>\n
Let the areas of the four sectors be $A_1, A_2, A_3, A_4$, and their corresponding central angles be $\\theta_1, \\theta_2, \\theta_3, \\theta_4$.
\nThe proportions of the areas are given as $A_1 : A_2 : A_3 : A_4 = 1 : 2 : 3 : 4$.
\nSince the areas are proportional to the angles, the ratio of the angles is also $\\theta_1 : \\theta_2 : \\theta_3 : \\theta_4 = 1 : 2 : 3 : 4$.<\/p>\n
The sum of the angles of the sectors in a circle is $360^\\circ$.
\nSo, $\\theta_1 + \\theta_2 + \\theta_3 + \\theta_4 = 360^\\circ$.<\/p>\n
The total proportion is $1 + 2 + 3 + 4 = 10$.
\nThe angles can be found by dividing the total angle ($360^\\circ$) according to the proportions:
\n$\\theta_1 = \\frac{1}{\\text{Total Proportion}} \\times 360^\\circ = \\frac{1}{10} \\times 360^\\circ = 36^\\circ$.
\n$\\theta_2 = \\frac{2}{10} \\times 360^\\circ = \\frac{1}{5} \\times 360^\\circ = 72^\\circ$.
\n$\\theta_3 = \\frac{3}{10} \\times 360^\\circ = \\frac{3}{10} \\times 360^\\circ = 108^\\circ$.
\n$\\theta_4 = \\frac{4}{10} \\times 360^\\circ = \\frac{2}{5} \\times 360^\\circ = 144^\\circ$.<\/p>\n
The four angles are $36^\\circ, 72^\\circ, 108^\\circ, 144^\\circ$.
\nCheck the sum: $36 + 72 + 108 + 144 = 108 + 108 + 144 = 216 + 144 = 360^\\circ$. The sum is correct.<\/p>\n