Similar Triangles

Similar Figures Similar figures have the same shape, but different size. The angles and side lengths will correspond just as they did in congruent figures, BUT… The corresponding side lengths will not be congruent. They will be proportional The corresponding angles will remain congruent

Similar Polygons S R Q P S R P Q To produce similar polygons, a dilation must have occurred (along with other possible transformations as well) Write down which vertices in quadrilateral ABCD correspond to which vertices in quadrilateral PQRS. The arrow “→” means “corresponds to.” A → B → C → D → The corresponding angles REMAIN CONGRUENT!! A  B  C  D  S R Q P R S P Q

Similar Polygons SP RS QR PQ = the scale factor RSAB 12 To produce similar polygons, a dilation must have occurred The side lengths will be a new size, but they will be proportional according to the scale factor. Write down the corresponding side lengths AB → BC → CD→ DA → The corresponding side lengths are proportional set up a fraction 4 2 3 5 12 6 9 15 RS SP QR PQ Image side length pre-image side length = the scale factor RSAB 12 4 = = 3 (the scale factor) QRDA 15 5 PQCD 6 2 SPBC 9 3 ALL corresponding side lengths need to be proportional = 3 = = 3 = = = 3

Are the two figures similar? Measure parts of both figures. How do we use these numbers to determine if the figures are similar? Find the scale factor. 8 ÷ 24 = 1/3 3 ÷ 9 = 1/3 Since the scale factor is the same for both sets of sides, the figures are similar. 9 cm 24 cm 3 cm 8 cm

What sequence of transformations changed the red figure into the brown figure? 9 cm It could have been a 90˚ clockwise rotation followed by a dilation with a scale factor of 1/3. Is it possible that it could have been a dilation with a scale factor of 1/3 then a 90˚ clockwise rotation? 24 cm 3 cm 8 cm

Similar Triangles Similar triangles have the same shape, but different size. You can use the relationships between corresponding parts of similar triangles to solve measurement problems.

The two triangles in the diagram are similar. To find the object’s height, you need to measure three distances and use similar triangles. What distances do you think we should measure? Person’s height Object’s distance to mirror. Person’s distance to mirror.

What do you do next? Once you have these three measurements, how do you find the height of the traffic light? Set up a proportion and solve for the missing height. 160 cm 600 cm 200 cm

Set up and solve the proportion Height of object = Height of person Distance of object to mirror Distance of person to mirror. x = 160 600 200 200x = 600(160) 200x = 96,000 x = 480 The height of the traffic light is 480 cm. 160 cm 600 cm 200 cm