Ratios and Proportions Similar Polygons

Ratio A ratio is a comparison of two quantities using division. A ratio of quantities a and b can be expressed as a to b, a:b, or a/b, where b ≠ 0. Ratios are usually expressed in simplest form.

Examples Sixteen students went on a week-long hiking trip. They brought with them 320 specially baked, protein-rich, cookies. What is the ratio of cookies to students?

Examples Sixteen students went on a week-long hiking trip. They brought with them 320 specially baked, protein-rich, cookies. What is the ratio of cookies to students? 320 cookies : 16 students 320 cookies/16 students : 16 students/16 students 20 cookies : 1 student

Extended Ratios Extended ratios can be used to compare three or more quantities. a:b:c means that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c, and the ratio of the first and last quantities is a:c.

Examples In a triangle, the ratio of the measures of the sides is 2:2:3 and the perimeter is 392 inches. Find the length of the longest side of the triangle.

Examples In a triangle, the ratio of the measures of the sides is 2:2:3 and the perimeter is 392 inches. Find the length of the longest side of the triangle. 2x + 2x + 3x = 392 7x = 392 x = 56 3(56) = 168

Proportions A proportion is an equation that says two ratios are equal Extremes are on the outside, means are on the inside; a:b = c:d

Examples Nathaniel is searching for a four-leaf clover in a field. He finds 2 four-leaf clovers during the first 12 minutes of his search. If Nathaniel spends a total of 180 minutes searching in the field, predict the number of four-leaf clovers Nathaniel will find.

Examples Nathaniel is searching for a four-leaf clover in a field. He finds 2 four-leaf clovers during the first 12 minutes of his search. If Nathaniel spends a total of 180 minutes searching in the field, predict the number of four-leaf clovers Nathaniel will find. 2/12 = x/180 360/12 = x x = 30

Equivalent Proportions Proportions will be equivalent as long as they have identical cross products.

Similar Polygons Two polygons are similar if and only if their corresponding angles are congruent and corresponding side lengths are proportional.

Examples In the diagram, NPQR ~ UVST. List all pairs of congruent angles, and write a proportion that relates the corresponding sides.

Examples In the diagram, NPQR ~ UVST. List all pairs of congruent angles, and write a proportion that relates the corresponding sides. N ≅ U P ≅ V Q ≅ S R ≅ T NP/NR = UV/UT

Scale Factor Scale factor is the ratio of the lengths of the corresponding sides of two similar polygons.

Examples Determine whether the triangles are similar. If so, write the similarity statement and scale factor. Explain your reasoning.

Examples Determine whether the triangles are similar. If so, write the similarity statement and scale factor. Explain your reasoning. Yes, they are similar NQP ~ RST s.f. – 5/4

Examples Find the value of each variable if △JLM ~ △QST.

Examples Find the value of each variable if △JLM ~ △QST. 4/2 = (3y - 2)/5 20/2 = 3y - 2 12 = 3y y = 4

Perimeters of Similar Polygons If two polygons are similar, then their perimeters are proportional to the scale factor between them.

Examples If MNPQ ~ XYZW, find the scale factor of MNPQ to XYZW and the perimeter of each polygon.

Examples If MNPQ ~ XYZW, find the scale factor of MNPQ to XYZW and the perimeter of each polygon. sf – 8/4 = 2 Perimeter of MNPQ = 34 Perimeter of XYZW = 17