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question:The points B(1, 1), I(2, 4) and G(5, 1) are plotted in the standard rectangular coordinate system to form triangle BIG. Triangle BIG is translated five units to the left and two units upward to triangle B'I'G', in such a way that B' is the image of B, I' is the image of I, and G' is the image of G. What is the midpoint of segment B'G'? Express your answer as an ordered pair.
answer:To translate a point 5 units to the left and 2 units upward, we subtract 5 from the x-coordinate and add 2 to the y-coordinate. The coordinates of B' are (1-5, 1+2) = (-4, 3), and the coordinates of G' are (5-5, 1+2) = (0, 3). The midpoint of segment B'G' is the average of the x-coordinates and the average of the y-coordinates, so the midpoint is left(frac{-4+0}{2}, frac{3+3}{2}right) = left(frac{-4}{2}, frac{6}{2}right) = boxed{(-2, 3)}. The answer is: (-2,3)
question:Zeoland the fine for speeding is 16 for each mile per hour the driver is traveling over the posted speed limit. In Zeoland, Jed was fined 256 for speeding on a road with a posted speed limit of x mph. Jed was fined for traveling at what speed in miles per hour? If we know the answer to the above question is 66, what is the value of unknown variable x?
answer:We know that the fine for speeding is 16 for each mile per hour the driver is traveling over the posted speed limit. Jed was fined 256, so he was traveling at 256 / 16 = 16 mph over the posted speed limit. Let's represent the posted speed limit as x mph. Jed was fined for traveling at x + 16 mph. We are given that Jed was fined for traveling at 66 mph, so we can write: x + 16 = 66. Solving for x, we get: x = 50. The value of x is 50. 50 The answer is: 50
question:James needs to get a new pair of glasses. His frames cost x and the lenses cost 500. Insurance will cover 80% of the cost of lenses and he has a 50 off coupon for frames. Everything costs 250. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the cost of the frames. Let's break down the information given: Cost of the frames: x Cost of the lenses: 500 Insurance coverage for lenses: 80% Coupon for frames: 50 off Total cost of frames and lenses: 250 We can set up the equation as follows: Cost of the frames + (Cost of the lenses - Insurance coverage for lenses) - Coupon for frames = Total cost of frames and lenses x + (500 - 0.8 * 500) - 50 = 250 Let's simplify and solve for x: x + (500 - 400) - 50 = 250 x + 100 - 50 = 250 x + 50 = 250 To isolate x, we subtract 50 from both sides of the equation: x + 50 - 50 = 250 - 50 x = 200 The value of x is 200. 200 The answer is: 200
question:right triangle PQR, we have angle Q = angle R and QR = 6sqrt{2}. What is the area of triangle PQR?
answer:Since angle Q = angle R, triangle PQR is an isosceles right triangle. Let x be the length of one of the legs. Then the length of the hypotenuse is xsqrt{2}, and by the Pythagorean theorem, we have [x^2 + x^2 = (xsqrt{2})^2.] Simplifying, we get 2x^2 = 2x^2, which is always true. Therefore, any positive value of x will satisfy the conditions of the problem. The area of triangle PQR is frac{1}{2}(text{base})(text{height}) = frac{1}{2}(x)(x) = frac{1}{2}x^2. Since the value of x does not affect the area, we can choose any positive value for x. For example, if we choose x = 6, then the area is frac{1}{2}(6)^2 = 18 square units. Therefore, the area of triangle PQR is boxed{18} square units. The answer is: 18