EXEHCRCE 1 posints \( A(3.8) \), B \( (-6,4) \) nt \( C(a,-7) \) 1. Calenier a pour que le linnghe Afri wit rockingte ewn 06 3. Soit. \( P \) le symetrigue de B par mopputt a M G. Geluler lea coordonnees de \( D \) 4. Quethe est In Hature du quandrlatóre ADCD? On justiflera la réponce sama effuct wer le moindre calcal 5. Calculer l'aire du quarinilatere \( A B C D \) ot win perimetere (on donnera ce resist sout bis forme \( a \sqrt{b}, a \) ef bentiers. b le plus petit. possible) 6. a) Développer, réduire et ordonner \( (z-6)(4 z+19) \).

Let's break down the problem step by step. ### 1. Find \( a \) such that the line \( AB \) is parallel to the line \( AC \). **Known Points:** - \( A(3, 8) \) - \( B(-6, 4) \) - \( C(a, -7) \) **Step 1: Calculate the slope of line \( AB \).** The slope \( m_{AB} \) is given by the formula: \[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 8}{-6 - 3} = \frac{-4}{-9} = \frac{4}{9} \] **Step 2: Calculate the slope of line \( AC \).** The slope \( m_{AC} \) is given by: \[ m_{AC} = \frac{-7 - 8}{a - 3} = \frac{-15}{a - 3} \] **Step 3: Set the slopes equal for parallel lines.** Since \( AB \) is parallel to \( AC \): \[ \frac{4}{9} = \frac{-15}{a - 3} \] **Step 4: Cross-multiply and solve for \( a \).** \[ 4(a - 3) = -135 \] \[ 4a - 12 = -135 \] \[ 4a = -123 \] \[ a = -\frac{123}{4} = -30.75 \] ### 2. Find the coordinates of point \( D \), the symmetric of \( B \) with respect to point \( M \). **Step 1: Find the midpoint \( M \) of segment \( AB \).** The coordinates of \( M \) are given by: \[ M\left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = M\left(\frac{3 + (-6)}{2}, \frac{8 + 4}{2}\right) = M\left(-\frac{3}{2}, 6\right) \] **Step 2: Find the coordinates of point \( D \).** The coordinates of \( D \) can be found using the midpoint formula: \[ M = \left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right) \] Setting up the equations: \[ -\frac{3}{2} = \frac{-6 + x_D}{2} \quad \text{and} \quad 6 = \frac{4 + y_D}{2} \] **Solving for \( x_D \) and \( y_D \):** 1. For \( x_D \): \[ -\frac{3}{2} = \frac{-6 + x_D}{2} \implies -3 = -6 + x_D \implies x_D = 3 \] 2. For \( y_D \): \[ 6 = \frac{4 + y_D}{2} \implies 12 = 4 + y_D \implies y_D = 8 \] Thus, \( D(3, 8) \). ### 3. Determine the nature of quadrilateral \( ABCD \). **Step 1: Check if \( A, B, C, D \) are vertices of a parallelogram.** To check if \( ABCD \) is a parallelogram, we can check if the midpoints of the diagonals \( AC \) and \( BD \) are the same. **Midpoint of \( AC \):** \[ M_{AC} = \left(\frac{3 + a}{2}, \frac{8 + (-7)}{2}\right) = \left(\frac{3 + a}{2}, \frac{1}{2}\right) \] **Midpoint of \( BD \):** \[ M_{BD} = \left(\frac{-6 + 3}{2}, \frac{4 + 8}{2}\right) = \left(-\frac{3}{2}, 6\right) \] Since the midpoints are not equal, \( ABCD \) is not a parallelogram. ### 4. Calculate the area of quadrilateral \( ABCD \) and its perimeter. **Step 1: Use the shoelace formula for area.** The area \( A \) of quadrilateral \( ABCD \) can be calculated using the formula: \[ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \] Substituting the coordinates: - \( A(3, 8) \) - \( B(-6, 4) \) - \( C(-30.75, -7) \) - \( D(3, 8) \) Calculating the area: \[ A = \frac{1}{2} \left| 3 \cdot 4 + (-6) \cdot (-7) + (-30.75) \cdot 8 + 3 \cdot 8 - (8 \cdot (-6) + 4 \cdot (-30.75) + (-7) \cdot 3 + 8 \cdot 3) \right| \] **Step 2: Calculate the perimeter.** The perimeter \( P \) is the sum of the lengths of the sides: \[ P = AB + BC + CD + DA \] Using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] ### 5. Develop, reduce, and order \( (z-6)(4z+19) \). **Step 1: Expand the expression.** \[ (z - 6)(4z + 19) = z \cdot 4z + z \cdot 19 - 6 \cdot 4z - 6 \cdot 19 \] \[ = 4z^2 + 19z - 24z - 114 \] \[ = 4z^2 - 5z - 114 \] Now, let's calculate the area and perimeter of quadrilateral \( ABCD \) and provide the final results. ### Final Calculations I will now calculate the area and perimeter using the shoelace formula and distance formula. Calculate the value by following steps: - step0: Calculate: \(3\times 4+\left(-6\right)\left(-7\right)+\left(-30\right)\) - step1: Remove the parentheses: \(3\times 4-6\left(-7\right)-30\) - step2: Multiply the numbers: \(12-6\left(-7\right)-30\) - step3: Multiply the numbers: \(12+42-30\) - step4: Calculate: \(24\) Calculate or simplify the expression \( (3*4 + (-6)*(-7) + (-30.75)*8 + 3*8) - (8*(-6) + 4*(-30.75) + (-7)*3 + 8*3) \). Calculate the value by following steps: - step0: Calculate: \(\left(3\times 4+\left(-6\right)\left(-7\right)+\left(-30.75\right)\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step1: Remove the parentheses: \(\left(3\times 4-6\left(-7\right)-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step2: Multiply the numbers: \(\left(12-6\left(-7\right)-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step3: Multiply the numbers: \(\left(12+42-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step4: Multiply the numbers: \(\left(12+42-246+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step5: Multiply the numbers: \(\left(12+42-246+24\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step6: Calculate: \(\left(-168\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)\) - step7: Remove the parentheses: \(-168-\left(8\left(-6\right)+4\left(-30.75\right)-7\times 3+8\times 3\right)\) - step8: Multiply the numbers: \(-168-\left(-48+4\left(-30.75\right)-7\times 3+8\times 3\right)\) - step9: Multiply the numbers: \(-168-\left(-48-123-7\times 3+8\times 3\right)\) - step10: Multiply the numbers: \(-168-\left(-48-123-21+8\times 3\right)\) - step11: Multiply the numbers: \(-168-\left(-48-123-21+24\right)\) - step12: Calculate: \(-168-\left(-168\right)\) - step13: Remove the parentheses: \(-168+168\) - step14: Add the numbers: \(0\) Calculate or simplify the expression \( \sqrt((3 - (-6))^2 + (8 - 4)^2) + \sqrt((-6 - (-30.75))^2 + (4 - (-7))^2) + \sqrt((-30.75 - 3)^2 + (-7 - 8)^2) + \sqrt((3 - 3)^2 + (8 - 8)^2) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{\left(3-3\right)^{2}+\left(8-8\right)^{2}}\) - step1: Subtract the terms: \(\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{\left(3-3\right)^{2}+0^{2}}\) - step2: Subtract the terms: \(\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step3: Remove the parentheses: \(\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step4: Remove the parentheses: \(\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step5: Remove the parentheses: \(\sqrt{\left(3+6\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step6: Add the numbers: \(\sqrt{9^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step7: Subtract the numbers: \(\sqrt{9^{2}+4^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step8: Add the numbers: \(\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step9: Add the numbers: \(\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step10: Subtract the numbers: \(\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step11: Subtract the numbers: \(\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step12: Add the numbers: \(\sqrt{97}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step13: Convert the expressions: \(\sqrt{97}+\sqrt{\left(\frac{99}{4}\right)^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step14: Add the numbers: \(\sqrt{97}+\sqrt{\frac{11737}{16}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step15: Simplify the root: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step16: Convert the expressions: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\left(-\frac{135}{4}\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}\) - step17: Add the numbers: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\frac{21825}{16}}+\sqrt{0^{2}+0^{2}}\) - step18: Simplify the root: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0^{2}+0^{2}}\) - step19: Calculate: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0+0^{2}}\) - step20: Calculate: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0+0}\) - step21: Remove 0: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0}\) - step22: Simplify the root: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+0\) - step23: Remove 0: \(\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}\) - step24: Reduce fractions to a common denominator: \(\frac{\sqrt{97}\times 4}{4}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}\) - step25: Transform the expression: \(\frac{\sqrt{97}\times 4+11\sqrt{97}+15\sqrt{97}}{4}\) - step26: Reorder the terms: \(\frac{4\sqrt{97}+11\sqrt{97}+15\sqrt{97}}{4}\) - step27: Add the numbers: \(\frac{30\sqrt{97}}{4}\) - step28: Reduce the fraction: \(\frac{15\sqrt{97}}{2}\) ### Summary of Calculations 1. **Finding \( a \)**: - We found \( a = -30.75 \) such that line \( AB \) is parallel to line \( AC \). 2. **Coordinates of Point \( D \)**: - The coordinates of point \( D \) are \( D(3, 8) \). 3. **Nature of Quadrilateral \( ABCD \)**: - The quadrilateral \( ABCD \) is not a parallelogram since the midpoints of the diagonals \( AC \) and \( BD \) are not equal. 4. **Area of Quadrilateral \( ABCD \)**: - The area calculated using the shoelace formula is \( 0 \). This indicates that points \( A, B, C, D \) are collinear. 5. **Perimeter of Quadrilateral \( ABCD \)**: - The perimeter is calculated as \( \frac{15\sqrt{97}}{2} \). 6. **Expansion of \( (z-6)(4z+19) \)**: - The expanded form is \( 4z^2 - 5z - 114 \). ### Final Results - \( a = -30.75 \) - Coordinates of \( D = (3, 8) \) - Area of quadrilateral \( ABCD = 0 \) - Perimeter of quadrilateral \( ABCD = \frac{15\sqrt{97}}{2} \) - Expanded form of \( (z-6)(4z+19) = 4z^2 - 5z - 114 \)