7-5 Parts of Similar Triangles. Find x.
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1 Find x. 1. By AA Similarity, the given two triangles are similar. Additionally, we see the segments marked x and 10 are medians because they intersect the opposite side at its midpoint. Theorem 7.10 states that if two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides. Therefore, esolutions Manual - Powered by Cognero Page 1
2 3. VISION A cat that is 10 inches tall forms a retinal image that is 7 millimeters tall. If and the distance from the pupil to the retina is 25 millimeters, how far away from your pupil is the cat? If two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides. Let the altitude from B meet AE at L and CD at K. Since triangle ABE and DBC are similar, triangle ALB and triangle DKB are similar. We know that BK = 25 mm, DK = 3.5 mm and AL = 5 inches. Use the Proportionality Theorem. Substitute in given values and solve for LB. 3.5LB = 2750 LB mm LB 35.7 in. The cat is 35.7 in. away from your pupil. esolutions Manual - Powered by Cognero Page 2
3 Find the value of each variable. 5. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. Find x. 7. By AA Similarity, the given two triangles are similar. Theorem 7.8 states that if two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides.we know that the sides marked 15 and 7.5 are altitudes because they are perpendicular to the side opposite a vertex of the triangle. Therefore, x = 8.5 esolutions Manual - Powered by Cognero Page 3
4 9. By AA Similarity, the given two triangles are similar.we know that the segments labeled x and 8 are altitudes because they are perpendicular to the side opposite a vertex. theorem 7.8 states that if two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides. Therefore, CCSS SENSE-MAKING Find the value of each variable. 11. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. esolutions Manual - Powered by Cognero Page 4
5 13. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. 15. ALGEBRA If and are altitudes, AB = 9, AD = 4x 8, JK = 21, and JM = 5x + 3, find x. Form a proportion. Substitute. Solve for x. esolutions Manual - Powered by Cognero Page 5
6 17. If is an altitude of is an altitude of RT = 5, RQ = 4, QY = 6, and YX = 2, find XV. If two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides. Substitute. esolutions Manual - Powered by Cognero Page 6
7 19. PROOF Write a two-column proof of Theorem Theorem 7.10 states that if two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides. Since we know that, we can use congruent corresponding angles and proportional sides to work towards getting, Since we already have and, we can use the given that and are medians to prove that, through segment addition postulate, definition of median, and substitution. Once this this established, you can prove that and, consequently, corresponding parts are proportional in these two triangles. Given: is a median of is a median of Prove: Proof: Statements (Reasons) 1. is a median of is a median of (Given) 2. CD = DB; TU = US (Def. of median) 3. (Def. of ) 4. CB = CD + DB; TS = TU + US (Seg. Add. Post.) 5. (Subst.) 6. (Subst.) 7. (Subst.) 8. (Def. of triangles) 9. (SAS Similarity) 10. (Def. of ) esolutions Manual - Powered by Cognero Page 7
8 ALGEBRA Find x. 21. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. 23. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. esolutions Manual - Powered by Cognero Page 8
9 CCSS ARGUMENTS Write a two-column proof. 25. Theorem 7.11 Given: bisects ACB. By construction, Prove: Theorem 7.11 states that an angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. Since you have a line ( ) that is parallel to one side of the triangle and divides the other sides of the triangle into two parts, by the Triangle Proportionality theorem, we know that. This looks very similar to the conclusion, except we need to replace EC with AC. Since, then and. From the given information, we can also conclude that (How?). Therefore,. This means that is an isosceles triangle and. Now, we can replace EC with AC in. Proof: Statements (Reasons) 1. bisects ; By construction, (Given) 2. ( Prop. Thm.) 3. (Def. of Angle Bisector) 4. (Alt. Int. angle Thm.) 5. (Corr. angle Post.) 6. (Trans. Prop.) 7. (Conv. of Isos. Thm.) 8. EC = AC (Def. of segs.) 9. (Subst.) esolutions Manual - Powered by Cognero Page 9
10 PROOF Write a two-column proof. 27. Given: and are angle bisectors. Prove: This proof is based on the idea that, if you can prove, then you can make a proportional statement regarding corresponding sides. So, since, we already know that. Therefore, we just need one more pair of congruent corresponding angles. We are given that and are angle bisectors so we can state that and. We also know that, so you can also prove that their half angles must also be congruent. Then, by AA Similarity, you can prove, and consequently,. Proof: Statements (Reasons) 1. and are angle bisectors. (Given) 2., (Def of s) 3., (Def. angle bisector) 4., (Angle Add. Thm.) 5., (Subst.) 6. (Subst.) 7. (Div.) 8. (def of congruent angles) 9. (AA Similarity) 10. (Def of ) esolutions Manual - Powered by Cognero Page 10
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