Leaving Cert Mathematics Trigonometry: The planned supports for a build

Question

The planned supports for a building's roof form scalene triangles of different sizes.

(i) Explain what is meant by a scalene triangle.

A scalene triangle is a type of triangle in which all three sides have different lengths, which consequently means that all three angles are also different. No sides or angles are equal, setting it apart from other types of triangles such as isosceles or equilateral triangles.

(ii) Find the length of $[FG]$.

To find the length of $[FG]$, we can use the sine rule, which states that in any triangle, the ratio of a side to the sine of its opposite angle is constant. We know that:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Here, we can assign $|CD| = 9$ m, $\angle C = 60^\circ$, and $|EF| = [FG]$. Thus, we can write:
$$\frac{|FG|}{\sin(\angle D)} = \frac{9}{\sin(\angle C)}$$
This will allow us to calculate the length.

(iii) Find the length of $[BD]$, correct to three decimal places.

Using the triangle's dimensions and applying the cosine rule, we can say:
$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$
Where:
- $c = BD$
- $a = 7$, $b = 8$
Calculating, we find:
$$[BD] = \sqrt{7^2 + 8^2 - 2 \cdot 7 \cdot 8\cdot \cos(60^\circ)}$$
After calculation, we round the result to three decimal places.

(iv) The centre of the enlargement is $O$. Find the distance from $O$ to point $B$.

The distance can be calculated using the properties of the triangles after determining the coordinates of points involved. The coordinates derived from the problem's dimensions should guide us, ensuring we use the distance formula:
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

(v) A condition of the planning is that the height of the point $G$ above the horizontal line $BF$ cannot exceed 11.6 m. Does the plan meet this condition? Justify your answer by calculation.

First, we determine the height of point $G$. Through geometric considerations and the known dimensions, we can find the height which can be represented as:
$$Height = G_y - BF_y$$
We then compare this height to the maximum height capacity of 11.6 m to verify whether it meets the conditions.

The planned supports for a building's roof form scalene triangles of different sizes - Leaving Cert Mathematics - Question (a) - 2012

Worked Solution & Example Answer:The planned supports for a building's roof form scalene triangles of different sizes - Leaving Cert Mathematics - Question (a) - 2012

Explain what is meant by a scalene triangle.

Find the length of [FG].

Find the length of [BD], correct to three decimal places.

Find the distance from O to point B.

Does the plan meet this condition? Justify your answer by calculation.

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