This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — To solve this problem, we will use the distance formula between two points (x_1, y_1) and (x_2, y_2): D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) Part 1: Show by calculation that HJ = JK Step 1: Calculate the length of HJ using coordinates H(-1, 6) and J(7, 8). HJ = sqrt((7 - (-1))^2 + (8 - 6)^2) HJ = sqrt((7 + 1)^2 + (2)^2) HJ = sqrt((8)^2 + 4) HJ = sqrt(64 + 4) HJ = sqrt(68) Step 2: Calculate the length of JK using coordinates J(7, 8) and K(5, 0). JK = sqrt((5 - 7)^2 + (0 - 8)^2) JK = sqrt((-2)^2 + (-8)^2) JK = sqrt(4 + 64) JK = sqrt(68) Step 3: Compare the lengths of HJ and JK. Since HJ = sqrt(68) and JK = sqrt(68), we can conclude that HJ = JK. Part 2: If it is further given that HL = LK, what type of quadrilateral is HJKL? Motivate your answer. Step 4: Calculate the length of HL using coordinates H(-1, 6) and L(-7, -6). HL = sqrt((-7 - (-1))^2 + (-6 - 6)^2) HL = sqrt((-7 + 1)^2 + (-12)^2) HL = sqrt((-6)^2 + 144) HL = sqrt(36 + 144) HL = sqrt(180) Step 5: Calculate the length of LK using coordinates L(-7, -6) and K(5, 0). LK = sqrt((5 - (-7))^2 + (0 - (-6))^2) LK = sqrt((5 + 7)^2 + (6)^2) LK = sqrt((12)^2 + 36) LK = sqrt(144 + 36) LK = sqrt(180) Step 6: Determine the type of quadrilateral HJKL. From our calculations, we have: HJ = sqrt(68) JK = sqrt(68) HL = sqrt(180) LK = sqrt(180) We observe that HJ = JK and HL = LK. Also, HJ ≠ HL (since sqrt(68) ≠ sqrt(180)). A quadrilateral with two distinct pairs of equal-length adjacent sides is a kite. The type of quadrilateral is a kite. Motivation: A kite is defined as a quadrilateral where two pairs of equal-length sides are adjacent to each other. In HJKL, the adjacent sides HJ and JK are equal, and the adjacent sides HL and LK are equal. Drop the next question.