A rectangular plot of land required 192 meters of fencing material to cover it. If one side is thrice the other, what is the area of the plot of land? A. 1342 sqm B. 432 sqm C. 6912 sqm D. 1728sqm Let's #learnwithlyqa!
Let the measure of angle A be \( x \) degrees. The complement of angle A is \( 90 - x \) degrees. According to the problem, angle A is 15 degrees more than twice its complement. This gives us the equation: \[ x = 2(90 - x) + 15 \] Let's solve for \( x \): \[ x = 180 - 2x + 15 \] \[ x = 195 - 2x \] \[ x + 2x = 195 \] \[ 3x = 195 \] \[ x = \frac{195}{3} \] \[ x = 65 \] So, the measure of angle A is 65 degrees.
Let the measure of angle B be \( y \) degrees. The complement of angle B is \( 90 - y \) degrees, and the supplement of angle B is \( 180 - y \) degrees. According to the problem, the complement of angle B is 16 degrees less than half of its supplement. This gives us the equation: \[ 90 - y = \frac{1}{2}(180 - y) - 16 \] Let's solve for \( y \): \[ 90 - y = 90 - \frac{y}{2} - 16 \] Simplify the right side of the equation: \[ 90 - y = 74 - \frac{y}{2} \] To eliminate the fraction, multiply every term by 2: \[ 2(90 - y) = 2(74 - \frac{y}{2}) \] \[ 180 - 2y = 148 - y \] Combine like terms: \[ 180 - 148 = 2y - y \] \[ 32 = y \] So, the measure of angle B is 32 degrees.
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Let the measure of the first angle be \( x \) degrees. The second angle is twice the first, so it is \( 2x \) degrees. The third angle is thrice the first, so it is \( 3x \) degrees. Let the measure of the fourth angle be \( y \) degrees. The sum of the angles in any quadrilateral is \( 360 \) degrees. Therefore, we have: \[ x + 2x + 3x + y = 360 \] Simplifying, we get: \[ 6x + y = 360 \] Since we are only given the relationship between the first three angles, we need to make an assumption about the fourth angle in order to solve for \( x \). If we assume the fourth angle is equal to the first angle (i.e., \( y = x \)), we can solve the equation: \[ 6x + x = 360 \] \[ 7x = 360 \] \[ x = \frac{360}{7} \] \[ x \approx 51.43 \] So, the measure of the first angle is approximately \( 51.43 \) degrees. If the assumption about the fourth angle being equal to the first angle isn't correct, we'd need more information to determine the exact measure of the first angle.