b) \( \mathrm{A}(5 ; 1), \mathrm{B}(1 ; 3) \) and \( \mathrm{C}(1 ;-2) \) c) \( \mathrm{A}(-2 ;-3), \mathrm{B}(-4 ; 1) \) and \( \mathrm{C}(4 ; 5) \) d) \( \mathrm{A}(0 ; 0), \mathrm{B}(\sqrt{3} ; 1) \) and \( \mathrm{C}(\sqrt{3} ;-1) \) e) \( \mathrm{A}(2 ;-1), \mathrm{B}(-3 ; 4) \) and \( \mathrm{C}(4 ; 5) \) f) \( \mathrm{A}(0 ; 0), \mathrm{B}(p ; q) \) and \( \mathrm{C}(-q ; p) \)

To analyze the points provided in each part (b to f), we will determine if the points A, B, and C are collinear. Points are collinear if the area of the triangle formed by them is zero. The area can be calculated using the determinant formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of points A, B, and C respectively. Let's calculate the area for each set of points. ### Part b: \( A(5, 1), B(1, 3), C(1, -2) \) Using the coordinates: - \( A(5, 1) \) → \( x_1 = 5, y_1 = 1 \) - \( B(1, 3) \) → \( x_2 = 1, y_2 = 3 \) - \( C(1, -2) \) → \( x_3 = 1, y_3 = -2 \) Calculating the area: \[ \text{Area} = \frac{1}{2} \left| 5(3 - (-2)) + 1(-2 - 1) + 1(1 - 3) \right| \] ### Part c: \( A(-2, -3), B(-4, 1), C(4, 5) \) Using the coordinates: - \( A(-2, -3) \) → \( x_1 = -2, y_1 = -3 \) - \( B(-4, 1) \) → \( x_2 = -4, y_2 = 1 \) - \( C(4, 5) \) → \( x_3 = 4, y_3 = 5 \) Calculating the area: \[ \text{Area} = \frac{1}{2} \left| -2(1 - 5) + (-4)(5 - (-3)) + 4(-3 - 1) \right| \] ### Part d: \( A(0, 0), B(\sqrt{3}, 1), C(\sqrt{3}, -1) \) Using the coordinates: - \( A(0, 0) \) → \( x_1 = 0, y_1 = 0 \) - \( B(\sqrt{3}, 1) \) → \( x_2 = \sqrt{3}, y_2 = 1 \) - \( C(\sqrt{3}, -1) \) → \( x_3 = \sqrt{3}, y_3 = -1 \) Calculating the area: \[ \text{Area} = \frac{1}{2} \left| 0(1 - (-1)) + \sqrt{3}(-1 - 0) + \sqrt{3}(0 - 1) \right| \] ### Part e: \( A(2, -1), B(-3, 4), C(4, 5) \) Using the coordinates: - \( A(2, -1) \) → \( x_1 = 2, y_1 = -1 \) - \( B(-3, 4) \) → \( x_2 = -3, y_2 = 4 \) - \( C(4, 5) \) → \( x_3 = 4, y_3 = 5 \) Calculating the area: \[ \text{Area} = \frac{1}{2} \left| 2(4 - 5) + (-3)(5 - (-1)) + 4(-1 - 4) \right| \] ### Part f: \( A(0, 0), B(p, q), C(-q, p) \) Using the coordinates: - \( A(0, 0) \) → \( x_1 = 0, y_1 = 0 \) - \( B(p, q) \) → \( x_2 = p, y_2 = q \) - \( C(-q, p) \) → \( x_3 = -q, y_3 = p \) Calculating the area: \[ \text{Area} = \frac{1}{2} \left| 0(q - p) + p(p - 0) + (-q)(0 - q) \right| \] Now, I will calculate the areas for parts b to f. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|5\left(3-\left(-2\right)\right)+1\times \left(-2-1\right)+1\times \left(1-3\right)\right|\) - step1: Remove the parentheses: \(\frac{1}{2}\left|5\left(3+2\right)+1\times \left(-2-1\right)+1\times \left(1-3\right)\right|\) - step2: Add the numbers: \(\frac{1}{2}\left|5\times 5+1\times \left(-2-1\right)+1\times \left(1-3\right)\right|\) - step3: Subtract the numbers: \(\frac{1}{2}\left|5\times 5+1\times \left(-3\right)+1\times \left(1-3\right)\right|\) - step4: Subtract the numbers: \(\frac{1}{2}\left|5\times 5+1\times \left(-3\right)+1\times \left(-2\right)\right|\) - step5: Multiply the numbers: \(\frac{1}{2}\left|25+1\times \left(-3\right)+1\times \left(-2\right)\right|\) - step6: Calculate: \(\frac{1}{2}\left|25-3+1\times \left(-2\right)\right|\) - step7: Calculate: \(\frac{1}{2}\left|25-3-2\right|\) - step8: Subtract the numbers: \(\frac{1}{2}\left|20\right|\) - step9: Calculate the absolute value: \(\frac{1}{2}\times 20\) - step10: Reduce the numbers: \(1\times 10\) - step11: Simplify: \(10\) Calculate or simplify the expression \( 1/2 * | -2 * (1 - 5) + (-4) * (5 - (-3)) + 4 * (-3 - 1) | \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|-2\left(1-5\right)+\left(-4\right)\left(5-\left(-3\right)\right)+4\left(-3-1\right)\right|\) - step1: Remove the parentheses: \(\frac{1}{2}\left|-2\left(1-5\right)+\left(-4\right)\left(5+3\right)+4\left(-3-1\right)\right|\) - step2: Subtract the numbers: \(\frac{1}{2}\left|-2\left(-4\right)+\left(-4\right)\left(5+3\right)+4\left(-3-1\right)\right|\) - step3: Remove the parentheses: \(\frac{1}{2}\left|-2\left(-4\right)-4\left(5+3\right)+4\left(-3-1\right)\right|\) - step4: Add the numbers: \(\frac{1}{2}\left|-2\left(-4\right)-4\times 8+4\left(-3-1\right)\right|\) - step5: Subtract the numbers: \(\frac{1}{2}\left|-2\left(-4\right)-4\times 8+4\left(-4\right)\right|\) - step6: Multiply the numbers: \(\frac{1}{2}\left|8-4\times 8+4\left(-4\right)\right|\) - step7: Multiply the numbers: \(\frac{1}{2}\left|8-32+4\left(-4\right)\right|\) - step8: Multiply the numbers: \(\frac{1}{2}\left|8-32-16\right|\) - step9: Subtract the numbers: \(\frac{1}{2}\left|-40\right|\) - step10: Calculate the absolute value: \(\frac{1}{2}\times 40\) - step11: Reduce the numbers: \(1\times 20\) - step12: Simplify: \(20\) Calculate or simplify the expression \( 1/2 * | 0 * (q - p) + p * (p - 0) + (-q) * (0 - q) | \). Simplify the expression by following steps: - step0: Solution: \(\frac{1}{2}\left|0\times \left(q-p\right)+p\left(p-0\right)+\left(-q\right)\left(0-q\right)\right|\) - step1: Remove the parentheses: \(\frac{1}{2}\left|0\times \left(q-p\right)+p\left(p-0\right)-q\left(0-q\right)\right|\) - step2: Multiply the numbers: \(\frac{1}{2}\left|0+p\left(p-0\right)-q\left(0-q\right)\right|\) - step3: Remove 0: \(\frac{1}{2}\left|0+p\times p-q\left(0-q\right)\right|\) - step4: Remove 0: \(\frac{1}{2}\left|0+p\times p-q\left(-q\right)\right|\) - step5: Multiply the terms: \(\frac{1}{2}\left|0+p^{2}-q\left(-q\right)\right|\) - step6: Multiply the terms: \(\frac{1}{2}\left|0+p^{2}+q^{2}\right|\) - step7: Remove 0: \(\frac{1}{2}\left|p^{2}+q^{2}\right|\) - step8: Calculate the absolute value: \(\frac{1}{2}\left(p^{2}+q^{2}\right)\) - step9: Apply the distributive property: \(\frac{1}{2}p^{2}+\frac{1}{2}q^{2}\) Calculate or simplify the expression \( 1/2 * | 0 * (1 - (-1)) + \sqrt(3) * (-1 - 0) + \sqrt(3) * (0 - 1) | \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|0\times \left(1-\left(-1\right)\right)+\sqrt{3}\times \left(-1-0\right)+\sqrt{3}\times \left(0-1\right)\right|\) - step1: Multiply: \(\frac{1}{2}\left|0+\sqrt{3}\times \left(-1-0\right)+\sqrt{3}\times \left(0-1\right)\right|\) - step2: Remove 0: \(\frac{1}{2}\left|0+\sqrt{3}\times \left(-1\right)+\sqrt{3}\times \left(0-1\right)\right|\) - step3: Remove 0: \(\frac{1}{2}\left|0+\sqrt{3}\times \left(-1\right)+\sqrt{3}\times \left(-1\right)\right|\) - step4: Multiply: \(\frac{1}{2}\left|0-\sqrt{3}+\sqrt{3}\times \left(-1\right)\right|\) - step5: Multiply: \(\frac{1}{2}\left|0-\sqrt{3}-\sqrt{3}\right|\) - step6: Remove 0: \(\frac{1}{2}\left|-\sqrt{3}-\sqrt{3}\right|\) - step7: Subtract the numbers: \(\frac{1}{2}\left|-2\sqrt{3}\right|\) - step8: Calculate the absolute value: \(\frac{1}{2}\times 2\sqrt{3}\) - step9: Reduce the numbers: \(1\times \sqrt{3}\) - step10: Simplify: \(\sqrt{3}\) Calculate or simplify the expression \( 1/2 * | 2 * (4 - 5) + (-3) * (5 - (-1)) + 4 * (-1 - 4) | \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|2\left(4-5\right)+\left(-3\right)\left(5-\left(-1\right)\right)+4\left(-1-4\right)\right|\) - step1: Remove the parentheses: \(\frac{1}{2}\left|2\left(4-5\right)+\left(-3\right)\left(5+1\right)+4\left(-1-4\right)\right|\) - step2: Subtract the numbers: \(\frac{1}{2}\left|2\left(-1\right)+\left(-3\right)\left(5+1\right)+4\left(-1-4\right)\right|\) - step3: Remove the parentheses: \(\frac{1}{2}\left|2\left(-1\right)-3\left(5+1\right)+4\left(-1-4\right)\right|\) - step4: Add the numbers: \(\frac{1}{2}\left|2\left(-1\right)-3\times 6+4\left(-1-4\right)\right|\) - step5: Subtract the numbers: \(\frac{1}{2}\left|2\left(-1\right)-3\times 6+4\left(-5\right)\right|\) - step6: Simplify: \(\frac{1}{2}\left|-2-3\times 6+4\left(-5\right)\right|\) - step7: Multiply the numbers: \(\frac{1}{2}\left|-2-18+4\left(-5\right)\right|\) - step8: Multiply the numbers: \(\frac{1}{2}\left|-2-18-20\right|\) - step9: Subtract the numbers: \(\frac{1}{2}\left|-40\right|\) - step10: Calculate the absolute value: \(\frac{1}{2}\times 40\) - step11: Reduce the numbers: \(1\times 20\) - step12: Simplify: \(20\) Here are the calculated areas for each set of points: ### Part b: \( A(5, 1), B(1, 3), C(1, -2) \) \[ \text{Area} = 10 \] Since the area is not zero, points A, B, and C are **not collinear**. ### Part c: \( A(-2, -3), B(-4, 1), C(4, 5) \) \[ \text{Area} = 20 \] Since the area is not zero, points A, B, and C are **not collinear**. ### Part d: \( A(0, 0), B(\sqrt{3}, 1), C(\sqrt{3}, -1) \) \[ \text{Area} = \sqrt{3} \] Since the area is not zero, points A, B, and C are **not collinear**. ### Part e: \( A(2, -1), B(-3, 4), C(4, 5) \) \[ \text{Area} = 20 \] Since the area is not zero, points A, B, and C are **not collinear**. ### Part f: \( A(0, 0), B(p, q), C(-q, p) \) \[ \text{Area} = \frac{1}{2}p^{2} + \frac{1}{2}q^{2} \] This expression is zero only if \( p = 0 \) and \( q = 0 \). Therefore, points A, B, and C are **not collinear** unless both \( p \) and \( q \) are zero. ### Summary: - **Part b**: Not collinear - **Part c**: Not collinear - **Part d**: Not collinear - **Part e**: Not collinear - **Part f**: Not collinear (unless \( p = 0 \) and \( q = 0 \))