Parts of Similar Triangles

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Presentation transcript:

Parts of Similar Triangles Similarity Transformations Scale Drawings and Models

Altitudes If two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of the corresponding sides.

Angle Bisectors If two triangles are similar, the lengths of corresponding angle bisectors are proportional to the lengths of the corresponding sides.

Medians If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of the corresponding sides.

Examples Find the value of x.

Examples Find the value of x. 16/20 = x/15 240/20 = x 12 = x

Examples Find the value of x.

Examples Find the value of x. x/13.5 = 3/9 x = 40.5/9 x = 4.5

Triangle Angle Bisector An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

Examples Find the value of x.

Examples Find the value of x. 4/6 = x/13 52/6 = x 8 2/3 = x

Examples Find the value of x.

Examples Find the value of x. x/14 = 20-x/11 11x = 280 – 14x 25x = 280

Similarity Transformations A dilation is a transformation that enlarges or reduces the original figure proportionally. Dilations are performed with respect to a fixed point called the center of dilation. The scale factor of a dilation describes the extent of the dilation and is the ratio of a length

Enlargement A dilation with a scale factor greater than 1 produces an enlargement, or an image that is larger than the original figure.

Reduction A dilation with a scale factor between 0 and 1 produces a reduction, an image that is smaller than the original figure.

Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation.

Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. Enlargement; 5/4

Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation.

Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. Reduction; 1/3

Verify Similarity You can verify that a dilation produces a similar figure by comparing corresponding sides and angles.

examples Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. Original: A(2, 3), B(0, 1), C(3, 0) Image: D(4, 6), F(0, 2), G(6, 0)

examples Original: A(2, 3), B(0, 1), C(3, 0) Image: D(4, 6), F(0, 2), G(6, 0)

examples Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. Original: H(0 ,0), J(6, 0), K(6, 4), L(0, 4) Image: W(0 ,0), X(3, 0), Y(3, 2), Z(0, 2)

examples Original: H(0 ,0), J(6, 0), K(6, 4), L(0, 4) Image: W(0 ,0), X(3, 0), Y(3, 2), Z(0, 2)

Scale Drawings and Models A scale model or scale drawing is an object or drawing with lengths proportional to the object it represents. The scale is the ratio of a length on the model to the actual length of the object being modeled.