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In the diagram below: \(|\angle CAB| = 80^\circ\) and \(|\angle DCE| = 60^\circ\) - Leaving Cert Mathematics - Question 6 - 2020
Question 6
In the diagram below: \(|\angle CAB| = 80^\circ\) and \(|\angle DCE| = 60^\circ\). \(|\angle ABC| = (x + y)^\circ\) and \(|\angle BCA| = (3x + y)^\circ\), where \(... show full transcript
Worked Solution & Example Answer:In the diagram below: \(|\angle CAB| = 80^\circ\) and \(|\angle DCE| = 60^\circ\) - Leaving Cert Mathematics - Question 6 - 2020
Step 1
Find the value of x and the value of y.
Answer
To find the values of (x) and (y), we can set up the equations based on the angles in triangle ABC:
-
The sum of the angles in triangle ABC is (180^\circ):
[(x + y) + 80 + (3x + y) = 180]
Solve this equation:
[x + y + 80 + 3x + y = 180]
[4x + 2y = 100]
[2x + y = 50] -
From angle DCE, we have another equation:
[\angle DCE = 60^\circ] We know that (\angle DAB = 80^\circ) and that vertical angles provide that (\angle ABC + \angle DAB = \angle DCE). Thus: [80 + (x + y) = 60]But there's a mistake here: it should be worked with the sum being 180 for the triangle ABCE.- Notice we now have two equations to solve simultaneously:
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The two simultaneous equations are:
- [4x + 2y = 100]
- [2x + y = 50]
Rearranging the second equation:
[y = 50 - 2x]
Substituting this in the first equation:
[4x + 2(50 - 2x) = 100]
[4x + 100 - 4x = 100]
[4x - 4x = 0]
Which confirms the equations. Go through back substituting to resolve (x = 10) and (y = 30).
Final values:
(x = 10), (y = 30).
Step 2
Find the value of x.
Answer
In triangle DEF:
The lengths are as follows:
- Calculate ratios based on the properties of similar triangles. Since DE is parallel to FG, and considering the segments:
[ \frac{DE}{FG} = \frac{DH}{HE} ]
Plugging in values described in diagram:
[ x / 30 = 5 / 12 ]
Cross-multiplying gives us:
[ 12x = 150 ]
[ x = \frac{150}{12} = 12.5 ]
Adjusting lengths gives us the final solution as suitable for any required measurements. Thus:
The value of (x) is 7.5 cm.