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Show by calculation that HJ = JK. If it is further given that HL = LK, what type of quadrilateral is HJKL? Motivate your answer.

Mathematics

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HJ = JK

Step-by-step solution

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To solve this problem, we will use the distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):

D=(x2−x1)2+(y2−y1)2D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Part 1· 3 steps

Show by calculation that HJ = JK

  1. Calculate the length of HJ using coordinates H(−1,6-1, 6) and J(7,87, 8).

    HJ=(7−(−1))2+(8−6)2HJ = \sqrt{(7 - (-1))^2 + (8 - 6)^2} HJ=(7+1)2+(2)2HJ = \sqrt{(7 + 1)^2 + (2)^2} HJ=(8)2+4HJ = \sqrt{(8)^2 + 4} HJ=64+4HJ = \sqrt{64 + 4} HJ=68HJ = \sqrt{68}

  2. Calculate the length of JK using coordinates J($7, 8$) and K($5, 0$).

    JK=(5−7)2+(0−8)2JK = \sqrt{(5 - 7)^2 + (0 - 8)^2} JK=(−2)2+(−8)2JK = \sqrt{(-2)^2 + (-8)^2} JK=4+64JK = \sqrt{4 + 64} JK=68JK = \sqrt{68}

  3. Compare the lengths of HJ and JK.

    Since HJ=68HJ = \sqrt{68} and JK=68JK = \sqrt{68}, we can conclude that HJ=JKHJ = JK.

Part 2· 3 steps

If it is further given that HL = LK, what type of quadrilateral is HJKL? Motivate your answer.

  1. Calculate the length of HL using coordinates H(−1,6-1, 6) and L(−7,−6-7, -6).

    HL=(−7−(−1))2+(−6−6)2HL = \sqrt{(-7 - (-1))^2 + (-6 - 6)^2} HL=(−7+1)2+(−12)2HL = \sqrt{(-7 + 1)^2 + (-12)^2} HL=(−6)2+144HL = \sqrt{(-6)^2 + 144} HL=36+144HL = \sqrt{36 + 144} HL=180HL = \sqrt{180}

  2. Calculate the length of LK using coordinates L(−7,−6-7, -6) and K(5,05, 0).

    LK=(5−(−7))2+(0−(−6))2LK = \sqrt{(5 - (-7))^2 + (0 - (-6))^2} LK=(5+7)2+(6)2LK = \sqrt{(5 + 7)^2 + (6)^2} LK=(12)2+36LK = \sqrt{(12)^2 + 36} LK=144+36LK = \sqrt{144 + 36} LK=180LK = \sqrt{180}

  3. Determine the type of quadrilateral HJKL.

    From our calculations, we have:
    HJ=68HJ = \sqrt{68}
    JK=68JK = \sqrt{68}
    HL=180HL = \sqrt{180}
    LK=180LK = \sqrt{180}

    We observe that HJ=JKHJ = JK and HL=LKHL = LK. Also, HJ≠HLHJ \neq HL (since 68≠180\sqrt{68} \neq \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.

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