Jse calculus to find the area of the triangle with the vertices \( 0,5),(2,-2) \), and \( (5,1) \) Area between curves =

To find the area of the triangle with the vertices \( (0,5) \), \( (2,-2) \), and \( (5,1) \), we can use the formula for the area of a triangle given its vertices. The formula for the area of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Given the vertices \( (0,5) \), \( (2,-2) \), and \( (5,1) \), we can substitute these values into the formula to find the area of the triangle. Let's calculate the area using the formula. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left|0\times \left(-2-1\right)+2\left(1-5\right)+5\left(5-\left(-2\right)\right)\right|\) - step1: Multiply: \(\frac{1}{2}\left|0+2\left(1-5\right)+5\left(5-\left(-2\right)\right)\right|\) - step2: Remove the parentheses: \(\frac{1}{2}\left|0+2\left(1-5\right)+5\left(5+2\right)\right|\) - step3: Subtract the numbers: \(\frac{1}{2}\left|0+2\left(-4\right)+5\left(5+2\right)\right|\) - step4: Add the numbers: \(\frac{1}{2}\left|0+2\left(-4\right)+5\times 7\right|\) - step5: Multiply the numbers: \(\frac{1}{2}\left|0-8+5\times 7\right|\) - step6: Multiply the numbers: \(\frac{1}{2}\left|0-8+35\right|\) - step7: Remove 0: \(\frac{1}{2}\left|-8+35\right|\) - step8: Add the numbers: \(\frac{1}{2}\left|27\right|\) - step9: Calculate the absolute value: \(\frac{1}{2}\times 27\) - step10: Multiply: \(\frac{27}{2}\) The area of the triangle with the vertices \( (0,5) \), \( (2,-2) \), and \( (5,1) \) is \( 13.5 \) square units.