Three friends distributed their total property. The ratio of total property, the share of the first friend, and the share of the second friend is \[24:7:8\]. Find the central angle of the sector, which represents the share of the third friend.
A. \[72\dfrac{1}{2}^\circ \]
B. \[135^\circ \]
C. \[105^\circ \]
D. \[67\dfrac{1}{2}^\circ \]

Question

Three friends distributed their total property. The ratio of total property, the share of the first friend, and the share of the second friend is \[24:7:8\]. Find the central angle of the sector, which represents the share of the third friend.
A. \[72\dfrac{1}{2}^\circ \]
B. \[135^\circ \]
C. \[105^\circ \]
D. \[67\dfrac{1}{2}^\circ \]

Vedantu Content Team · Accepted Answer

Hint: Here, we will first assume that the total property is $$24x$$, the share of the first friend be $$7x$$, and the share of the second friend be $$8x$$. Then we will find the share of the third friend by subtracting the sum of the share of the first and second friends from the total property. We know that the angle representing the total property is equal to the 360 degrees, we will use this to find the value of $$x$$ and then substitute back in the share of the third friend to the central angle.Complete step by step answer:We are given that the ratio of total property, the share of the first friend, and the share of the second friend is $$24:7:8$$.Let us assume that the total property is $$24x$$, the share of the first friend be $$7x$$, and the share of the second friend be $$8x$$.Then we will find the share of the third friend by subtracting the sum of the share of the first and second friend from the total property, we get$$   \Rightarrow 24x - \left( {7x + 8x} \right)   \\   \Rightarrow 24x - 15x   \\   \Rightarrow 9x   \\  $$We know that the angle representing the total property is equal to the 360 degrees, so we have$$ \Rightarrow 24x = 360^\circ $$Dividing the above equation by 24 on both sides, we get$$   \Rightarrow \dfrac{{24x}}{{24}} = \dfrac{{360^\circ }}{{24}}   \\   \Rightarrow x = 15   \\  $$Substituting the above value of $$x$$ in the share of the third friend to the central angle, we get$$   \Rightarrow 9 \times 15   \\   \Rightarrow 135^\circ    \\  $$Thus, the central angle is 135 degrees.Hence, option B is correct.Note: While solving this question, we need to remember the angle representing the total property equals 360 degrees, not 90 degrees, or else the answer will be wrong. The angle subtended by the sector to the center of the circle and the central angle of an arc length. We know that the length around the curved arc is what defines the sector.