In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC. F is a point on BC such that EF || AC. AB produced and DF produced meet in G. 10.1 Prove that: $ \frac{BC}{FC} = \frac{AC}{DA} $ 10.2 Prove that: $ \triangle BFE \parallel \triangle EDA $ 10.3 It is further given that EF = 2, BF = 3.5, and ED = 10, determine the length of: 10.3.1 AD 10.3.2 DC

NSC Mathematics Euclidean Geometry: In the diagram below, D and E ar

In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC - NSC Mathematics - Question 10 - 2017 - Paper 2

Question 10

In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC. F is a point on BC such that EF || AC. AB produced and DF produ... show full transcript

Worked Solution & Example Answer:In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC - NSC Mathematics - Question 10 - 2017 - Paper 2

Step 1

Prove that: $ \frac{BC}{FC} = \frac{AC}{DA} $

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Answer

To prove this, we can use the properties of similar triangles and parallel lines. Since DE is parallel to BC, by the basic proportionality theorem, we have:

$\frac{BA}{EA} = \frac{BC}{FC}$

By considering triangle ACD and applying the same theorem, we also get:

$\frac{CA}{DA} = \frac{AC}{DA}$

Thus, we can combine these ratios:

$\frac{BC}{FC} = \frac{AC}{DA}$

Step 2

Prove that: $ \triangle BFE \parallel \triangle EDA $

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Answer

Using the corresponding angles, we know:

Angle at F is equal to angle at D,
Angle at E is equal to angle at A.

Thus, the triangles are similar due to the Angle-Angle (AA) postulate, indicating that:

$\triangle BFE \parallel \triangle EDA$

Step 3

Determine the length of AD

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Answer

To find the length of AD, we can apply the similarity ratios obtained from the earlier parts of the question. Given that EF = 2, BF = 3.5, and ED = 10, we can set up the ratio:

$\frac{AD}{ED} = \frac{BF}{EF}$

Substituting the known values:

$\frac{AD}{10} = \frac{3.5}{2} \implies AD = 10 \times \frac{3.5}{2} = 17.5$

Step 4

Determine the length of DC

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Answer

Since DC is opposite to EF, we can utilize the length of EF to find DC. Given that EF = 2, and noting that AD is already calculated:

$DC = EF = 2$

In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC - NSC Mathematics - Question 10 - 2017 - Paper 2

Worked Solution & Example Answer:In the diagram below, D and E are points on sides AC and AB respectively of Δ ABC such that DE || BC - NSC Mathematics - Question 10 - 2017 - Paper 2

Prove that: \( \frac{BC}{FC} = \frac{AC}{DA} \)

Prove that: \( \triangle BFE \parallel \triangle EDA \)

Determine the length of AD

Determine the length of DC

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