Circle T has a center at (1, –2). Point B(–2, 2) is on the circle, line BU is a tangent line, and segment BN is a diameter of circle T.
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Determine the perimeter and area of ΔBUN. Round to the hundredths place, if necessary. For each drop-down, select the correct number.
perimeter of ΔBUN =
units area of ΔBUN =
square units
Given that BN is a diameter of circle T, it passes through the center of the circle at (1, -2). As a diameter, it also passes through the point B(-2, 2). Therefore, we can see that the slope of line BN is (2 - (-2))/(-2 - 1) = 4/-3 = -4/3.
Since BU is a tangent line, it is perpendicular to BN. Therefore, the slope of line BU is the negative reciprocal of -4/3, which is 3/4.
Now, we can find the equation of line BU. Point B(-2, 2) lies on line BU and we know the slope is 3/4. Therefore, the equation of line BU can be found using point-slope form: y - 2 = (3/4)(x + 2). Simplifying this gives y = (3/4)x + 1.
Next, we can find the coordinates of point U by finding the intersection of lines BU (y = (3/4)x + 1) and BN. Substituting the equation of line BU into the equation of line BN gives:
(3/4)x + 1 = -4/3(x - 1)
Solving for x gives x = -4. Substituting x = -4 back into the equation of line BU gives y = 3. Therefore, point U is (-4, 3).
Now, we can find the lengths of sides BU, UN, and BN using the distance formula:
Length of BU = sqrt((-4 - (-2))^2 + (3 - 2)^2) = sqrt(2^2 + 1^2) = sqrt(5)
Length of UN = sqrt((-4 - 1)^2 + (3 - (-2))^2) = sqrt(5^2 + 5^2) = sqrt(50) = 5sqrt(2)
Length of BN = 2*radius = 2*sqrt((1 - (-2))^2 + (-2 - 2)^2) = 2*sqrt(3^2 + 4^2) = 2*sqrt(9+16) = 2*sqrt(25) = 10
Therefore, the perimeter of triangle BUN = sqrt(5) + 5sqrt(2) + 10.
To find the area, we can use the distance formula to find the base of the triangle (base is UN) and the height of the triangle (height is the perpendicular distance from U to line BN).
Height = absolute value of (-4*(3 - 2) + 3*(1 + 2)) / sqrt(3^2 + 4^2) = 9 / 5 = 1.8
Area = (1/2) * base * height = (1/2) * 5sqrt(2) * 1.8 = 4.5sqrt(2)
Therefore, the perimeter of triangle BUN is approximately 18.24 units and the area of triangle BUN is approximately 6.36 square units.