SIMILARITY

Transcription

1 SIMILRITY In this section students focus on comparing geometric shapes. The begin b dilating shapes: enlarging them as one might on a cop machine. hen students compare the original and enlarged shapes closel, the discover that the shape of the figure remains eactl the same (this means the angle measures of the enlarged figure are equal to those of the original figure), but the size changes (the lengths of the sides increase). lthough the size changes, the lengths of the corresponding sides all have a constant ratio, known as the scale factor, or ratio of similarit. See the Math Notes boes in Lessons and 2.. for more information about dilations and similar figures. ample nlarge the figure at right from the origin b a factor of. Students used rubber bands to create a dilation (enlargement) of several shapes. e can do this using a grid and slope triangles. reate a right triangle so that the segment from the origin to point (2, ), is the hpotenuse, one leg lies on the positive -ais, and the other leg connects point to the endpoint of the first leg at (2, 0). This triangle is called a slope triangle since it represents the slope of the hpotenuse from (0, 0) to verte. dd two more slope triangles eactl like this one along the ra from (0, 0) through point as shown in the figure at right. Using three triangles creates an enlargement b a factor of and gives us the new point at (, ). Repeat this process for the other two vertices, forming a new slope triangle for each verte. This will give us new points at (, ) and at (, ). onnecting points,, and, we form a new triangle that is an enlargement of the original triangle b a factor of, as shown at left. Notice that the sides of the dilated triangle are parallel to the sides of the original triangle. Parent Guide with tra Practice 20 PM ducational Program. ll rights reserved. 2

2 hapter 2 ample 2 The two quadrilaterals at right are similar. hat parts are equal? an ou determine the lengths of an of the unlabeled sides? Since the quadrilaterals are similar, we know that all the corresponding angles have the same measure. This means that m = m, m = m, m = m, and m = m. In addition, the corresponding sides are proportional, which means the ratio of corresponding sides is a constant. To determine the ratio, we need to know the lengths of one pair of corresponding sides. From the diagram we see that corresponds to. Since these sides correspond, we can write =. Therefore, the ratio of similarit is, or 2. e can use this value to calculate the length of another side if we know the length of its corresponding side. = = = = = = = = = = = ample The pair of shapes at right is similar (F ~ UXYZ). Label the second figure correctl to reflect the similarit statement. ssume the second figure is drawn to scale. F Since the polgons are similar, this means that their corresponding angles have equal measure. hen we write a similarit statement, we write the letters so that the corresponding angles match up. the similarit statement, we must have m = m U, m = m, m = m, m = m X, m = m Y, and m F = m Z. The smaller figure is labeled at right. If it is difficult to tell which original angle corresponds to its enlargement or reduction, tr rotating the figures so that the have the same orientation. U X Z Y PM ducational Program. ll rights reserved. ore onnections Integrated II

3 Problems. op the figure below onto graph paper 2. reate a figure similar to the one and then enlarge it b a factor of 2. below with a scale factor of 0.. For each pair of similar figures below, determine the ratio of similarit for large:small. iagrams are drawn roughl to scale; ou can assume that sides that look longer are longer For each pair of similar figures, state the ratio of similarit. Then use it to calculate the value of. iagrams are drawn roughl to scale; ou can assume that sides that look longer are longer Parent Guide with tra Practice 20 PM ducational Program. ll rights reserved. 2

4 hapter 2. The shadow of a statue is 20 feet long, while the shadow of a student is ft long. If the student is ft tall, how tall is the statue? ach pair of figures below is similar. Use what ou know about similarit to solve for. iagrams are drawn roughl to scale; ou can assume that sides that look longer are longer Solve for the missing lengths in the pairs of similar figures below. 20. ~ PQR 2. JKLM ~ XYZ K R J P M L Q Y Z 0. X 22. STU ~ MNOP 2. ~ IS S T U O 2. 2 M N..2 P S I 2 20 PM ducational Program. ll rights reserved. ore onnections Integrated II

5 2. ~ FGHIJ 2. ~ H J I F G nswers ; = 2. 2 ; = 2. ; =. ; =. ; = ; =.. 0 ft. =. =. = 0.. = =. 2. = = =. 2. = Parent Guide with tra Practice 20 PM ducational Program. ll rights reserved. 2