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.
Since the triangle is isosceles we know that ∠CBA = ∠ACB and since the three angles in the... show full transcript
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
To find ∠ACB, we first apply the properties of isosceles triangles. Since ABC is isosceles with vertex A having an angle of 36°, we can say:
∠CBA = ∠ACB = x. Thus, the equation becomes:
x + x + 36° = 180°
This simplifies to:
2x = 144°
Therefore, solving for x gives:
∠ACB = 72°.
Step 2
On the diagram construct the bisector of ∠ABC. Show all construction lines clearly.
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Using a compass, place the point on line BA and draw arcs that intersect the lines BA and BC. Label these intersection points as E and F. From points E and F, draw arcs of equal radius that intersect inside the triangle. The intersection of these arcs gives the point G. Finally, draw a straight line from point A to point G, which is the bisector of ∠ABC.
Step 3
Mark in the point D where your bisector meets the line AC.
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Mark the point of intersection of the bisector line AG with line AC as point D.
Step 4
Calculate all angles in the triangle BCD and write them into the diagram.
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Answer
In triangle BCD, we know:
∠DCB = 72° (as calculated earlier)
∠CBD = 36° (since it is bisected)
To find ∠BDC, we use the angle sum property:
72° + 36° + ∠BDC = 180°
Thus,
∠BDC = 180° - 108° = 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, the triangle BCD is isosceles because it has two equal angles: ∠DCB and ∠BDC both equal to 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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The ratio can be written as:
|AC| : |BC| = \frac{95}{60} = 1.5833.
When rounded to three decimal places:
|AC| : |BC| = 1.583.
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