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In triangle, ABC, points D and F are on LINE AB and E is on AC such that DE parrerl to BC and EF parreell CD. If AF=9  and DF=3, then what is BD?

 Feb 26, 2023
 #1
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Since DE is parallel to BC, we can use the intercept theorem to find the length of DB. Let x be the length of DB, then:

(AD/x) = (AF/AB) [using the intercept theorem]

Since AF = 9, we have:

(AD/x) = 9/AB

Now, we need to find AB in terms of x. To do this, we will use the fact that EF is parallel to CD, which implies that triangles AEF and ACD are similar. Therefore, we have:

(AF/AC) = (AE/AD) = (EF/CD)

Substituting AF = 9 and EF = CD - DF, we get:

9/AC = AE/AD = (CD - 3)/CD

Simplifying this equation, we get:

CD = 3AC/(2AD - AE)

Now, we need to find AE in terms of x. To do this, we can use the fact that triangles ADE and ABC are similar. Therefore, we have:

(AE/AC) = (DE/BC) = (AD/AB)

Substituting DE = BC - x and AD/x = 9/AB, we get:

AE/AC = (BC - x)/(AB) = 9/AB

Simplifying this equation, we get:

AB^2 = 9AC(B + x)

Now, we can substitute CD and AB in terms of x in the above equation, and solve for x. We get:

x = 9/4

Therefore, the length of BD is 9/4.

 Feb 27, 2023
 #2
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We can use similar triangles to solve this problem.

Since DE is parallel to BC, we have ∠ECD = ∠ABC and ∠FDC = ∠ACB (alternate interior angles). Therefore, triangles ACD and DEF are similar.

Let x be the length of BD. Then, since AF = 9 and DF = 3, we have AB = AF + FB = 9 + x and BD = DF + FB = 3 + x.

Since triangles ACD and DEF are similar, we have

AC/CD = AD/DE

Substituting AC = AB - BC and CD = BD - BC, we get

(AB - BC)/(BD - BC) = AD/DE

Substituting AB = 9 + x and DE = BC, we get

(9 + x - BC)/(BD - BC) = AD/BC

Cross-multiplying, we get

AD = (9 + x - BC) × BC/(BD - BC)

Since triangles ABD and DFC are similar, we have

AD/DF = BD/FC

Substituting AD from the previous equation and DF = 3, we get

(9 + x - BC) × BC/(BD - BC) = BD/(BD - 3)

Cross-multiplying, we get

BD^2 - 6BD + 9 = BC × (9 + x)

Substituting BC = DE and using the fact that DE is parallel to BC, we have

BD^2 - 6BD + 9 = DE × (9 + x)

Substituting DE = BC and using the fact that EF is parallel to CD, we have

BD^2 - 6BD + 9 = EF × (BD - 3)

Substituting EF = CD - CF = BD - 3 and simplifying, we get

BD^2 - 12BD + 36 = 0

Factoring, we get

(BD - 6)^2 = 0

Therefore, BD = 6.

 

Thanks,
MyMorri Portal

 Feb 27, 2023

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