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The lengths of the sides of a flat triangular field ACB are, |AB| = 120 m, |BC| = 134 m and |AC| = 150 m - Leaving Cert Mathematics - Question 1 - 2014
Question 1
The lengths of the sides of a flat triangular field ACB are, |AB| = 120 m, |BC| = 134 m and |AC| = 150 m. (a) (i) Find ∠CBA. Give your answer, in degrees, correct ... show full transcript
Worked Solution & Example Answer:The lengths of the sides of a flat triangular field ACB are, |AB| = 120 m, |BC| = 134 m and |AC| = 150 m - Leaving Cert Mathematics - Question 1 - 2014
Step 1
Find ∠CBA. Give your answer, in degrees, correct to two decimal places.
Answer
To find ∠CBA, we can apply the Law of Cosines:
where:
- a = |AC| = 150 m
- b = |BC| = 134 m
- c = |AB| = 120 m
Substituting the values into the formula:
Now, to find angle B:
Thus, ∠CBA = 72.15° (to two decimal places).
Step 2
Find the area of the triangle ACB correct to the nearest whole number.
Answer
The area of triangle ACB can be calculated using the formula:
Using the values:
- a = |AB| = 120 m
- c = |BC| = 134 m
- B = 72.15°
Substituting our values into the formula gives:
Rounding to the nearest whole number, the area is approximately 7653 m².
Step 3
Explain why the three cables [EA], [EB] and [EC] are equal in length.
Answer
At the circumcentre D of the triangle ABC, the distance from D to each vertex (A, B, C) is equal because the circumcentre is the point where the perpendicular bisectors of the sides intersect, equidistant from all vertices.
Thus, each of the cables [EA], [EB], and [EC] shares the same length as they are each the hypotenuse of the right triangles formed by the segments AD, BD, and CD, respectively. According to the Pythagorean theorem, the lengths of these not only get established as equal but also signify the same distance to the mast from every vertex, confirming the lengths are equal.