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7.4: Corresponding Parts of Similar Figures

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    Identify matching side lengths and angle measures in similar figures.

    f-d_5e9b727f47bd4684a032eafdb97666f1904a13ebd45f10eb8802a84b+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.jpg
    Figure \(\PageIndex{1}\)

    Maria has an art project due for school. She decides to make a small replica of her bedroom. She begins by measuring the angles of her bedroom, then she draws similar figures. The larger figure represents her bedroom and the smaller figure represents her replica.

    f-d_9430380dab4a7758b24b8c4c42ebab5f69c461ee7001ed8497a899e7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Her bedroom door is located in the middle of \(\overline{IJ}\). For her replica, the door must be placed in the middle of the corresponding side. What side of the replica corresponds to \(\overline{IJ}\)?

    In this concept, you will learn about corresponding sides of similar figures.

    Corresponding Parts of Similar Figures

    f-d_8b34f510d01d38aebf0d1ca2866560ed65ba118231c436b90eece3aa+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Triangle ABC is similar to triangle DEF. This means that while they are the same shape, they aren’t the same size. In fact, there is a relationship between the corresponding parts of the triangle.

    The side lengths are corresponding even though they aren’t congruent. Use the symbol for similar ("\(\sim\)") to show the relationship between the corresponding sides of the two triangles.

    \(\overline{AB}\sim \overline{DE}\)

    \(\overline{BC}\sim \overline{EF}\)

    \(\overline{AC}\sim \overline{DF}\)

    Example \(\PageIndex{1}\)

    Earlier, you were given a problem about Maria and her bedroom.

    She is making a replica of her bedroom for her art class. She draws two similar figures to make sure that she is placing everything in the right position.

    f-d_9430380dab4a7758b24b8c4c42ebab5f69c461ee7001ed8497a899e7+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    She needs to place her bedroom door in the middle of the side that corresponds to \(\overline{IJ}\). What side corresponds to \(\overline{IJ}\)?

    First, rotate the second figure so that it looks like the first figure.

    Then, locate the side on the second figure that is in the position that \(\overline{IJ}\) is in the first figure.

    \(\overline{YZ}\)

    The answer is that \(\overline{YZ}\) corresponds to \(\overline{IJ}\). Maria needs to place her bedroom door in the middle of \(\overline{YZ}\) on the replica.

    Example \(\PageIndex{2}\)

    The figures below are similar. What side corresponds \(\overline{NO}\)?

    f-d_35adcfa8a46eacc3a3bebc362c1da0e5b0d79a6d69798e305b1c508c+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{5}\)

    Solution

    First, position the second figure so that it looks like the first figure.

    Then, identify the side in the second figure that is in the same position as \(\overline{NO}\) is in the first figure.

    \(\overline{QR}\)

    The answer is that \(\overline{QR}\) corresponds to \(\overline{NO}\).

    Example \(\PageIndex{3}\)

    \(\overline{AB}\sim _____\)

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    Figure \(\PageIndex{6}\)

    Solution

    First, look to see which side in the second triangle corresponds to the given segment.

    \(\overline{EF}\)

    Then, complete the problem.

    \(\overline{AB}\sim \overline{EF}\)

    The answer is that \(\overline{AB}\sim \overline{EF}\).

    Example \(\PageIndex{4}\)

    What makes the two squares similar if they both have four right angles?

    f-d_dcd7d42705fdf6a6a65f05919156260777bb0668ef284887c9b02f1a+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.jpg
    Figure \(\PageIndex{7}\)

    Solution

    First, check to see if the side lengths are the same.

    No

    Next, check to see if the shape is the same.

    Yes

    Then, state why the two squares are similar.

    Same shape, different sizes

    The answer is that the two squares are similar because they have the same shape, but different sizes.

    Example \(\PageIndex{5}\)

    In the triangle pair, are the two triangles similar or congruent?

    f-d_6bc62ac0ecc457dcf18c61d63f7581b028cba200e08fb93963e6edfa+IMAGE_THUMB_POSTCARD_TINY+IMAGE_THUMB_POSTCARD_TINY.jpgFigure \(\PageIndex{8}\)

    Solution

    First, check to see if the two triangles are the same size.

    No

    Next, check to see if the two triangles are the same shape.

    Yes

    Then, determine if the two triangles are similar or congruent.

    Similar

    The two triangles are similar.

    Review

    Use the following figures to answer each question.

    f-d_d6ac9211e7f3ea5788c6f3d3373d793afa1429025d267d6ae2c3efbf+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)
    1. Are these two triangles similar or congruent?
    2. How do you know?
    3. Which side is congruent to \(\overline{AB}\)?
    4. Which side is congruent to \(\overline{AC}\)?
    5. Which side is congruent to \(\overline{RS}\)?
    6. Which angle is congruent to angle A?
    7. Which angle is congruent to angle B?
    8. Which angle is congruent to angle C?
    f-d_1f0dab88064e7993cd372b0424e8909e0d2b1b418e70805b3b2d055c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)
    1. Are the two figures similar or congruent?
    2. Why?
    3. Which side is congruent to \(\overline{NO}\)?
    4. Which side is congruent to \(\overline{MN}\)?
    5. Which side is congruent to \(\overline{ST}\)?
    6. Which side is congruent to \(\overline{QT}\)?
    7. Which side is congruent to \(\overline{OP}\)?

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 9.17.

    Resources

    Vocabulary

    Term Definition
    Congruent Congruent figures are identical in size, shape and measure.
    Corresponding The corresponding sides between two triangles are sides in the same relative position.
    Similar Two figures are similar if they have the same shape, but not necessarily the same size.

    Additional Resources

    Video: Congruent and Similar Triangles

    Practice: Corresponding Parts of Similar Figures

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