Answer:
The value of x is 3
The value of m is 5.25
The value of a is 1.75
Step-by-step explanation:
I attached the picture with labels to can solve
In the 2 Δs ADE and ACF
∵ DE // CF
∴ m∠ADE = m∠ACF ⇒ corresponding angles
∴ m∠AED = m∠AFC ⇒ corresponding angles
→ By using the AA postulate of similarity
∴ The two triangles are similar
∴ Their corresponding sides have the same ratio
∴ [tex]\frac{AD}{AC}[/tex] = [tex]\frac{EA}{FA}[/tex] = [tex]\frac{DE}{CF}[/tex]
∵ AD = AB + BC + CD
∵ AB = 3, BC = 6, CD = x
∴ AD = 3 + 6 + x
∴ AD = 9 + x
∵ AC = AB + BC = 3 + 6
∴ AC = 9
∵ AE = AG + GF + FE
∵ AG = 4, GF = 8, FE = 4
∴ AE = 4 + 8 + 4
∴ AE = 16
∵ FA = AG + GF = 4 + 8
∴ FA = 12
∵ DE = 7 and CF = m
→ Substitute them in the ratio above
∴ [tex]\frac{9+x}{9}[/tex] = [tex]\frac{16}{12}[/tex] = [tex]\frac{7}{m}[/tex]
→ By using cross multiplication with the first 2 fractions
∵ 12 × (9 + x) = 9 × 16
∴ 12(9) + 12(x) = 144
∴ 108 + 12x = 144
→ Subtract 108 from both sides
∴ 108 - 108 + 12x = 144 - 108
∴ 12x = 36
→ Divide both sides by 12 to find x
∴ x = 3
→ By using cross multiplication with the second 2 fractions
∵ 16 × m = 12 × 7
∴ 16m = 84
→ Divide both sides by 16
∴ m = 5.25
In the 2 Δs ABG and ACF
∵ BG // CF
∴ m∠ABG = m∠ACF ⇒ Corresponding angles
∴ m∠AGB = m∠AFC ⇒ corresponding angles
→ By using the AA postulate of similarity
∴ The two triangles are similar
∴ Their corresponding sides have the same ratio
∴ [tex]\frac{AB}{AC}[/tex] = [tex]\frac{GA}{FA}[/tex] = [tex]\frac{BG}{CF}[/tex]
∵ AB = 3 and AC = 9
∵ GA = 4 and FA = 12
∵ BG = a and CF = m = 5.25
→ Substitute them in the ratio above
∴ [tex]\frac{3}{6}[/tex] = [tex]\frac{4}{12}[/tex] = [tex]\frac{a}{5.25}[/tex]
→ By using cross multiplication with the second 2 fractions
∵ 12 × a = 4 × 5.25
∴ 12a = 21
→ Divide both sides by 12 to find a
∴ a = 1.75
