The triangle ABC is isosceles, as shown - Junior Cycle Mathematics - Question 11 - 2014
Question 11
The triangle ABC is isosceles, as shown.
∠BAC = 36°.
(i) Calculate ∠ACB.
(ii) On the diagram construct the bisector of ∠ABC. Show all construction lines clearly.
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Worked Solution & Example Answer:The triangle ABC is isosceles, as shown - Junior Cycle Mathematics - Question 11 - 2014
Step 1
Calculate ∠ACB.
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Answer
Since triangle ABC is isosceles, we have:
∠CAB+∠ACB+∠CBA=180°
Here, ∠CBA = ∠ACB (both are equal in an isosceles triangle). Therefore, we can denote ∠ACB as x. Thus, we can write:
36°+x+x=180°
This simplifies to:
36°+2x=180°
Subtracting 36° from both sides gives:
2x=144°
Dividing both sides by 2 yields:
x=72°
Thus,
∠ACB=72°.
Step 2
On the diagram construct the bisector of ∠ABC. Show all construction lines clearly.
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Answer
Use a compass to mark an arc that crosses both sides of line AB and line AC.
Without changing the radius, mark arcs from the points where the first arc intersects line AB and line AC.
Label the new intersection points as E and F.
Draw a line from point C to where the two new arcs intersect, which is the bisector of ∠ABC.
Step 3
Mark in the point D where your bisector meets the line AC.
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Label the intersection of the bisector that you just drew with line AC as point D.
Step 4
Calculate all the angles in the triangle BCD and write them into the diagram.
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Answer
In triangle BCD:
We know ∠BCD = 72° (from part i).
∠CBD is bisected, so ∠CBD = 36°.
The angles in triangle BCD must add up to 180°:
∠BCD+∠CBD+∠BDC=180°
Thus:
72°+36°+∠BDC=180°
Therefore:
∠BDC=180°−108°=72°
Consequently:
∠BCD = 72°
∠BDC = 72°
Step 5
Can you conclude that the triangle BCD is also isosceles? Give a reason for your answer.
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Answer
Yes, triangle BCD is isosceles because it has two equal angles,
∠BCD = ∠BDC = 72°.
Step 6
Measure |AC| and |BC|.
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|AC| = 95 mm
|BC| = 60 mm.
Step 7
Calculate the ratio |AC|/|BC|, correct to three places of decimals.
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Answer
The ratio can be computed as follows:
∣BC∣∣AC∣=6095=1.5833ext(tothreedecimalplaces)
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