Henry has ridden 76 miles of a bike course. The course is 95 miles long. What percentage of the course has Henry ridden so far?

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• To prove that the quadrilateral ABCD, with vertices A(2,1), B(1,3), C(-5,0), and D(-4,-2), is a rectangle, we need to demonstrate that it has four right angles and opposite sides that are equal in length. First, calculate the slopes of the sides to verify that consecutive sides are perpendicular, implying that all angles are right angles. The slope of line segment AB is given by: m_AB = (y2 - y1) / (x2 - x1) = (3 - 1) / (1 - 2) = 2 / -1 = -2. Similarly, calculate the slopes of the other sides: m_BC = (y_C - y_B) / (x_C - x_B) = (0 - 3) / (-5 - 1) = -3 / -6 = 1/2, m_CD = (y_D - y_C) / (x_D - x_C) = (-2 - 0) / (-4 - -5) = -2 / 1 = -2, m_DA = (y_A - y_D) / (x_A - x_D) = (1 - (-2)) / (2 - (-4)) = 3 / 6 = 1/2. Notice that slopes m_AB and m_CD are both -2, indicating AB and CD are parallel. Similarly, slopes m_BC and m_DA are both 1/2, indicating BC and DA are parallel. Additionally, the slopes of AB and BC, and CD and DA are negative reciprocals, implying that AB is perpendicular to BC and CD is perpendicular to DA, thus confirming that all angles are right angles. Next, verify that opposite sides are of equal length using the distance formula: d = √((x2 - x1)² + (y2 - y1)²). The length of side AB is: d_AB = √((1 - 2)² + (3 - 1)²) = √((-1)² + (2)²) = √(1 + 4) = √5. Calculate the lengths of the other sides: d_BC = √((-5 - 1)² + (0 - 3)²) = √((-6)² + (-3)²) = √(36 + 9) = √45, d_CD = √((-4 - -5)² + (-2 - 0)²) = √((1)² + (-2)²) = √(1 + 4) = √5, d_DA = √((2 - -4)² + (1 - -2)²) = √((6)² + (3)²) = √(36 + 9) = √45. The lengths of AB and CD are both √5, and the lengths of BC and DA are both √45, confirming that opposite sides are equal. Since both pairs of opposite sides are parallel and equal in length, and all four angles are right angles, quadrilateral ABCD is indeed a rectangle. To graph the rectangle, plot the points on a coordinate plane and draw lines connecting them in order, from A to B, B to C, C to D, and D back to A. Use graphing tools or software for precise visualization and to provide graphical evidence to support the analytical proof.