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The diagram shows the triangle ABC - Junior Cycle Mathematics - Question 7 - 2015

Question 7

The diagram shows the triangle ABC. DE is parallel to BC. The sizes of some of the angles are shown. (a) Find the value of x. (b) Given that |AE| = 100, |AC| = 130... show full transcript

Worked Solution & Example Answer:The diagram shows the triangle ABC - Junior Cycle Mathematics - Question 7 - 2015

Step 1

Find the value of x.

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Answer

To find the value of x, we start by using the information provided about the angles in triangle ABC.

Since DE is parallel to BC, the alternate angles are equal. Therefore, we can set up the equation based on the angles in the triangle:

$2x + 3x + 70° = 180°$

Combining the terms gives us:

$5x + 70° = 180°$

Subtracting 70° from both sides, we find:

$5x = 110°$

Dividing by 5:

$x = 22°$

Step 2

Given that |AE| = 100, |AC| = 130, and |DE| = 74, find the value of |BC|.

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Answer

To find |BC|, we can use the concept of proportional triangles. Since DE is parallel to BC, the ratios of the segments are equal:

$\frac{|BC|}{|AC|} = \frac{|DE|}{|AE|}$

Substituting the known values:

$\frac{|BC|}{130} = \frac{74}{100}$

To isolate |BC|, we cross-multiply:

$100 |BC| = 130 \times 74$

Calculating the right side:

$100 |BC| = 9620$

Now, dividing both sides by 100 gives us:

$|BC| = 96.2$

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