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In the diagram, △ABC and △ACD are drawn - NSC Mathematics - Question 9 - 2017 - Paper 2
Question 9
In the diagram, △ABC and △ACD are drawn. F and G are points on sides AB and AC respectively such that AF = 3x, FB = 2x, AG = 12y and GC = 8y. H, E and K are points o... show full transcript
Worked Solution & Example Answer:In the diagram, △ABC and △ACD are drawn - NSC Mathematics - Question 9 - 2017 - Paper 2
Step 1
FG || BC
Answer
To prove that FG || BC, we can apply the concept of similar triangles.
From the geometric configuration, we know that:
- AF : AG = 3x : 12y
- FB : CG = 2x : 8y
This gives us the proportion: By the converse of the Basic Proportionality Theorem, since the line segments are proportional, it follows that FG || BC.
Step 2
AH / HK = AE / ED
Answer
Applying the properties of parallel lines and similar triangles, we see:
- Given that GH || CK, we can use the theorem of proportional segments.
- This leads us to the proportion: Thus, we can assert that the ratio of the segments AH to HK equals the ratio of AE to ED.
Step 3
If it is further given that AH = 15 and ED = 12, calculate the length of EK.
Answer
We know that:
- AH = 15 and ED = 12, with HK as the unknown variable.
Using the established ratio: To solve for EK:
- Calculate AE using the ratios from similar triangles.
- From the geometric relationships, if AE is consistent with known values, it is derived by the relation EK = ED - HK.
- Solve for EK based on previously calculated lengths.