A chord with a length of 3√ (2 + √2) cm contracts an arc whose degree measure is 135.
January 30, 2021 | education
| A chord with a length of 3√ (2 + √2) cm contracts an arc whose degree measure is 135. Find the area of a circular sector corresponding to this arc?
Let’s draw the radii OA and OB from the center of the circle O. The AOB triangle is isosceles, since ОА = ОВ = R.
Let’s use the cosine theorem for a triangle.
AB2 = R ^ 2 + R ^ 2 – 2 * R * R * CosAOB.
(3 * √ (2 + √2) ^ 2 = 2 * R ^ 2 – 2 * R ^ 2 * (-√2 / 2).
9 * (2 + √2) = 2 * R ^ 2 + R ^ 2 * √2 = R ^ 2 * (2 + √2).
9 = R ^ 2.
R = 3 cm.
Let’s define the area of the OAB sector.
S = (n * R² * α) / 360 = (n * 9 * 135) / 360 = n * 27/8 cm2.
Answer: The area of the sector is n * 27/8 cm2.
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