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question:the diagram, AB is a line segment. What is the value of x? [asy] draw((0,0)(10,0),black+linewidth(1)); draw((4,0)(4,8),black+linewidth(1)); draw((4,0)(3.5,0)(3.5,0.5)(4,0.5)cycle,black+linewidth(1)); draw((4,0)(9,7),black+linewidth(1)); label(A,(0,0),W); label(B,(10,0),E); label(x^circ,(4.75,2.25)); label(52^circ,(5.5,0.75)); [/asy]
answer:Given the diagram, we have a line segment AB and a perpendicular line CD intersecting AB at point C. There's also a line CE forming angles with CD and AB. We need to find the value of x, which is the measure of the angle between lines CD and CE. 1. We know that the sum of angles around a point is 360^circ. Specifically, for a straight line like AB, the sum of angles on either side of the line is 180^circ. 2. The angle ACD is a right angle, so it measures 90^circ. 3. The angle ECB is given as 52^circ. 4. Therefore, using the property that the sum of angles on a straight line is 180^circ, we have: [ angle ACD + angle DCE + angle ECB = 180^circ ] Substituting the known values, we get: [ 90^circ + x^circ + 52^circ = 180^circ ] 5. Solving for x, we rearrange the equation: [ x^circ = 180^circ - 90^circ - 52^circ ] 6. Performing the subtraction, we find: [ x^circ = 38^circ ] Therefore, the value of x is boxed{38}.
question:The expression ( x_{1} : x_{2} : x_{3} : ldots : x_{n} ) has a definite value only when parentheses are used to indicate the order of divisions. The expression can be rewritten in the form [ frac{x_{i_{1}} x_{i_{2}} ldots x_{i_{k}}}{x_{j_{1}} x_{j_{2}} ldots x_{j_{nk}}} ] where ( i_{1}, i_{2}, ldots, i_{k} ) and ( j_{1}, j_{2}, ldots, j_{nk} ) are any permutation of the indices ( 1, 2, ldots, n ) with ( i_{1} < i_{2} < ldots < i_{k} ) and ( j_{1} < j_{2} < ldots < j_{nk} ). For example, [ left(x_{1} : left(x_{2} : x_{3}right)right) : x_{4} = frac{x_{1} x_{3}}{x_{2} x_{4}}, quad left(left(x_{1} : x_{2}right) : x_{3}right) : x_{4} = frac{x_{1}}{x_{2} x_{3} x_{4}} ] How many distinct fractions of this form can be obtained by placing the parentheses in different ways?
answer:1. **Understanding the Expression:** The given expression is ( x_{1} : x_{2} : x_{3} : ldots : x_{n} ). To interpret this properly, we need to specify the order of divisions using parentheses. For example, the expression can be rewritten as: frac{x_{i_{1}} x_{i_{2}} ldots x_{i_{k}}}{x_{j_{1}} x_{j_{2}} ldots x_{j_{n-k}}} where (i_{1}, i_{2}, ldots, i_{k}) and (j_{1}, j_{2}, ldots, j_{n-k}) represent numbers sorted from 1 to (n), with (i_{1} < i_{2} < ldots < i_{k}) and (j_{1} < j_{2} < ldots < j_{n-k}). 2. **Counting for (n = 3):** For (n = 3), consider the following orders: begin{aligned} & left(x_{1} : x_{2}right) : x_{3} = frac{x_{1}}{left(x_{2} x_{3}right)} & x_{1} : left( x_{2} : x_{3} right) = frac{x_{1} x_{3}}{x_{2}} end{aligned} This gives two different outcomes. 3. **Counting for (n = 4):** For (n = 4): - **Parentheses placement variations:** 1. ( left( left( x_{1} : x_{2} right) : x_{3} right) : x_{4} ) 2. ( x_{1} : left( left( x_{2} : x_{3} right) : x_{4} right) ) 3. ( left( x_{1} : left( x_{2} : x_{3} right) right) : x_{4} ) 4. ( x_{1} : left( x_{2} : left( x_{3} : x_{4} right) right) ) 5. ( left( x_{1} : x_{2} right) : left( x_{3} : x_{4} right) ) 6. The same as the previous if we select different order, not listed separately because the result is the same. - **Calculating distinct fractions:** Each different ordering yields: frac{x_{1} x_{3}}{x_{2} x_{4}}, quad frac{x_{1} x_{4}}{x_{2} x_{3}}, quad frac{x_{1}}{x_{2} x_{3} x_{4}}, quad frac{x_{1} x_{3} x_{4}}{x_{2}} Since the order (like 3,1,2) may yield the same fraction as another order (like 1, 3, 2), thus giving us 4 distinct fractions. 4. **General Pattern/A Formula:** For a general ( n )-term sequence, every subsequent term can either be put into the numerator or the denominator. This results in (2^{n-2}) outcomes as: - (x_{1}) (always in the numerator) - (x_{2}) (always in the denominator) - (x_{3}, x_{4}, ldots, x_{n}) can be chosen by binary choices (in the numerator or denominator). Mathematically, this can be validated as the number of distinct proper parenthesized ways of placing (n-1) division operations on (n) elements is given by: frac{1}{n} binom{2n-2}{n-1} 5. **Conclusion:** There are (2^{n-2}) distinct ways to assign the elements (x_{3}, x_{4}, ldots, x_{n}) to the numerator and denominator in the given sequence. Thus, the number of different fractions obtainable by varying the order of operations in the expression ( x_{1} : x_{2} : x_{3} : ldots : x_{n} ) is (2^{n-2}). (boxed{2^{n-2}})
question:triangle ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively. If c^{2}sin A=5sin C, (a+c)^{2}=16+b^{2}, find the area of triangle ABC.
answer:Analysis: This problem involves using the Sine Law, Cosine Law, and the formula for the area of a triangle to solve for the triangle. The key is to use the Sine Law and Cosine Law to find the cosine of angle B. Step 1: Using the Sine Law and the given information, we have {c}^{2}a=5c, So, ac=5, Step 2: From (a+c)^{2}=16+{b}^{2}, we get, {a}^{2}+{c}^{2}-{b}^{2}=6, Step 3: Using the Cosine Law, we get cos B= frac{{a}^{2}+{c}^{2}-{b}^{2}}{2ac}= frac{6}{10}= frac{3}{5}, Step 4: So, sin B= sqrt{1-cos^2 B} = frac{4}{5}, Step 5: Now, we can find the area of the triangle using the formula S= frac{1}{2}acsin B = frac{1}{2}times5times frac{4}{5}=boxed{2}.
question:Find the least common multiple of 36 and 105.
answer:First, we factorize 36 and 105 into their prime factors: [ 36 = 2^2 cdot 3^2, quad 105 = 3^1 cdot 5^1 cdot 7^1. ] Next, we identify the highest power of each prime that appears in the factorizations: - For prime 2: highest power is 2^2 (from 36). - For prime 3: highest power is 3^2 (from 36). - For prime 5: highest power is 5^1 (from 105). - For prime 7: highest power is 7^1 (from 105). Now, multiply these highest powers together to find the LCM: [ text{lcm}[36, 105] = 2^2 cdot 3^2 cdot 5^1 cdot 7^1 = 4 cdot 9 cdot 5 cdot 7 = 1260. ] Thus, the least common multiple of 36 and 105 is: [ boxed{1260}. ]