In the figure below, E is the midpoint of BC, AD: DC=3:2 and AE intersect with BD at F. Given that AB=b and AC=c, express in terms of b and c: i) AE ii) BD. If BF=tBD and AF=sAE, where t and s are scalars, find the values of t and s. What is the ratio in which F divides AE.

Question

Asked on August 2, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-08-02T17:12:36.443Z

Huu hapa ni ufumbuzi wa swali lako: a) i) Kueleza AE kwa kutumia b na c. Step 1: Tafuta vekta BC. Tunajua BC = AC - AB. BC = c - b Step 2: Tafuta vekta BE. Kwa kuwa E ni katikati ya BC, BE = (1)/(2)BC. BE = (1)/(2)(c - b) Step 3: Tafuta vekta AE. Tunajua AE = AB + BE. AE = b + (1)/(2)(c - b) AE = b + (1)/(2)c - (1)/(2)b AE = (1)/(2)b + (1)/(2)c ii) Kueleza BD kwa kutumia b na c. Step 1: Tafuta vekta AD. Tunapewa AD : DC = 3 : 2. Hii inamaanisha D inagawanya AC kwa uwiano wa 3:2. AD = (3)/(3+2)AC = (3)/(5)AC AD = (3)/(5)c Step 2: Tafuta vekta BD. Tunajua BD = AD - AB. BD = (3)/(5)c - b BD = -b + (3)/(5)c b) Kutafuta thamani za t na s. Step 1: Eleza AF kwa kutumia s. Tunapewa AF = sAE. Tumia AE kutoka sehemu (a)(i). AF = s((1)/(2)b + (1)/(2)c) AF = (s)/(2)b + (s)/(2)c (Equation 1) Step 2: Eleza AF kwa kutumia t. Tunajua AF = AB + BF. Pia tunapewa BF = tBD. Tumia BD kutoka sehemu (a)(ii). AF = b + tBD AF = b + t(-b + (3)/(5)c) AF = b - tb + (3t)/(5)c AF = (1-t)b + (3t)/(5)c (Equation 2) Step 3: Linganisha coefficients za b na c kutoka Equation 1 na Equation 2. Kulinganisha coefficients za b: (s)/(2) = 1-t (Equation 3) Kulinganisha coefficients za c: (s)/(2) = (3t)/(5) (Equation 4) Step 4: Tatua milinganyo ya wakati mmoja kupata t na s. Kutoka Equation 3 na Equation 4, tunaweza kulinganisha upande wa kulia: 1-t = (3t)/(5) Panua pande zote mbili kwa 5: 5(1-t) = 3t 5 - 5t = 3t 5 = 8t t = (5)/(8) Badilisha thamani ya t katika Equation 4: (s)/(2) = (3)/(5) × (5)/(8) (s)/(2) = (3)/(8) s = 2 × (3)/(8) s = (6)/(8) = (3)/(4) t = (5)/(8), s = (3)/(4) c) Uwiano ambao F inagawanya AE. Step 1: Tumia thamani ya s kutoka sehemu (b). Tunajua AF = sAE. Kutoka sehemu (b), s = (3)/(4). AF = (3)/(4)AE Hii inamaanisha kwamba vekta AF ni (3)/(4) ya vekta AE. Step 2: Tafuta uwiano wa AF : FE. Ikiwa AF ni sehemu 3 kati ya sehemu 4 za AE, basi FE ni sehemu iliyobaki. FE = AE - AF = AE - (3)/(4)AE = (1)/(4)AE Kwa hiyo, uwiano wa AF : FE ni: AF : FE = (3)/(4)AE : (1)/(4)AE = 3 : 1 AF : FE = 3 : 1 Tuma swali linalofuata 📸

Accepted Answer · 2026-08-02T17:12:36.443Z

Huu hapa ni ufumbuzi wa swali lako: a) i) Kueleza AE kwa kutumia b na c. Step 1: Tafuta vekta BC. Tunajua BC = AC - AB. BC = c - b Step 2: Tafuta vekta BE. Kwa kuwa E ni katikati ya BC, BE = (1)/(2)BC. BE = (1)/(2)(c - b) Step 3: Tafuta vekta AE. Tunajua AE = AB + BE. AE = b + (1)/(2)(c - b) AE = b + (1)/(2)c - (1)/(2)b AE = (1)/(2)b + (1)/(2)c ii) Kueleza BD kwa kutumia b na c. Step 1: Tafuta vekta AD. Tunapewa AD : DC = 3 : 2. Hii inamaanisha D inagawanya AC kwa uwiano wa 3:2. AD = (3)/(3+2)AC = (3)/(5)AC AD = (3)/(5)c Step 2: Tafuta vekta BD. Tunajua BD = AD - AB. BD = (3)/(5)c - b BD = -b + (3)/(5)c b) Kutafuta thamani za t na s. Step 1: Eleza AF kwa kutumia s. Tunapewa AF = sAE. Tumia AE kutoka sehemu (a)(i). AF = s((1)/(2)b + (1)/(2)c) AF = (s)/(2)b + (s)/(2)c (Equation 1) Step 2: Eleza AF kwa kutumia t. Tunajua AF = AB + BF. Pia tunapewa BF = tBD. Tumia BD kutoka sehemu (a)(ii). AF = b + tBD AF = b + t(-b + (3)/(5)c) AF = b - tb + (3t)/(5)c AF = (1-t)b + (3t)/(5)c (Equation 2) Step 3: Linganisha coefficients za b na c kutoka Equation 1 na Equation 2. Kulinganisha coefficients za b: (s)/(2) = 1-t (Equation 3) Kulinganisha coefficients za c: (s)/(2) = (3t)/(5) (Equation 4) Step 4: Tatua milinganyo ya wakati mmoja kupata t na s. Kutoka Equation 3 na Equation 4, tunaweza kulinganisha upande wa kulia: 1-t = (3t)/(5) Panua pande zote mbili kwa 5: 5(1-t) = 3t 5 - 5t = 3t 5 = 8t t = (5)/(8) Badilisha thamani ya t katika Equation 4: (s)/(2) = (3)/(5) × (5)/(8) (s)/(2) = (3)/(8) s = 2 × (3)/(8) s = (6)/(8) = (3)/(4) t = (5)/(8), s = (3)/(4) c) Uwiano ambao F inagawanya AE. Step 1: Tumia thamani ya s kutoka sehemu (b). Tunajua AF = sAE. Kutoka sehemu (b), s = (3)/(4). AF = (3)/(4)AE Hii inamaanisha kwamba vekta AF ni (3)/(4) ya vekta AE. Step 2: Tafuta uwiano wa AF : FE. Ikiwa AF ni sehemu 3 kati ya sehemu 4 za AE, basi FE ni sehemu iliyobaki. FE = AE - AF = AE - (3)/(4)AE = (1)/(4)AE Kwa hiyo, uwiano wa AF : FE ni: AF : FE = (3)/(4)AE : (1)/(4)AE = 3 : 1 AF : FE = 3 : 1 Tuma swali linalofuata 📸

In the figure below, E is the midpoint of BC, AD: DC=3:2 and AE intersect with BD at F. Given that AB=b and AC=c, express in terms of b and c: i) AE ii) BD. If BF=tBD and AF=sAE, where t and s are scalars, find the values of t and s. What is the ratio in which F divides AE.

Need help with your own homework?

More Mathematics Questions

In the figure below, E is the midpoint of BC, AD: DC=3:2 and AE intersect with BD at F. Given that AB=b and AC=c, express in terms of b and c: i) AE ii) BD. If BF=tBD and AF=sAE, where t and s are scalars, find the values of t and s. What is the ratio in which F divides AE.

Need help with your own homework?

More Mathematics Questions