In the diagram, ABCD is a parallelogram with AB = 14 units. AD is produced to E such that AD:DE = 4:3. EB intersects DC in F. EB = 21 units. Calculate, with reasons, the length of FB.

Here are the solutions to your geometry problem. 9.1 Calculate, with reasons, the length of FB. Step 1: Identify similar triangles. Since ABCD is a parallelogram, AB is parallel to DC. This means AB is parallel to DF. Consider EDF and EAB: DEF = AEB (Common angle) EDF = EAB (Corresponding angles, since DF AB and AE is a transversal) DFE = ABE (Corresponding angles, since DF AB and EB is a transversal) Therefore, EDF EAB (AAA similarity). Step 2: Use the ratio of corresponding sides to find EF. Given AD : DE = 4 : 3. Let AD = 4x and DE = 3x. Then AE = AD + DE = 4x + 3x = 7x. The ratio of corresponding sides is (ED)/(EA) = (3x)/(7x) = (3)/(7). From the similarity, we have (EF)/(EB) = (ED)/(EA). (EF)/(21) = (3)/(7) EF = (3)/(7) × 21 EF = 9 units Step 3: Calculate FB. FB = EB - EF FB = 21 - 9 FB = 12 units 9.2 Prove, with reasons, that EDF ||| EAB. To prove that EDF is similar to EAB, we need to show that their corresponding angles are equal. Step 1: Identify equal angles. DEF = AEB (Common angle) Since ABCD is a parallelogram, AB DC. Therefore, AB DF. EDF = EAB (Corresponding angles, as AB DF and AE is a transversal) DFE = ABE (Corresponding angles, as AB DF and EB is a transversal) Step 2: Conclude similarity. Since all three corresponding angles are equal, by the Angle-Angle-Angle (AAA) similarity criterion: EDF ||| EAB 9.3 Calculate, with reasons, the length of FC. Step 1: Use the ratio of corresponding sides to find DF. From the similarity EDF EAB (proven in 9.2), we have: (DF)/(AB) = (ED)/(EA) We know (ED)/(EA) = (3)/(7) (from 9.1) and AB = 14 units. (DF)/(14) = (3)/(7) DF = (3)/(7) × 14 DF = 6 units Step 2: Calculate FC. Since ABCD is a parallelogram, opposite sides are equal in length. DC = AB = 14 units Now, we can find FC: FC = DC - DF FC = 14 - 6 FC = 8 units Send me the next one 📸