Question

A roof has a cross-section as shown in the diagram. Find the height ‘h’ of the roof.

Answer 1

Hint: Find the area of the triangle using the formula: $\text{A}=\dfrac{1}{2}\times \text{base}\times \text{height}$, where ‘A’ is the area of the triangle. Consider ‘XY’ as the base and ‘ZY’ as the height of the triangle XYZ and find its area given by: $\text{A}=\dfrac{1}{2}\times XY\times YZ$. Also, consider ‘XZ’ as the base and ‘YW’ as the height of the triangle XYZ and find its area given by: $\text{A}=\dfrac{1}{2}\times ZX\times YW$. Substitute all the given values of sides and equate the two areas with each other to find the value of ‘h’.

Complete step-by-step answer:
We know that, area of a triangle is given as: $\text{A}=\dfrac{1}{2}\times \text{base}\times \text{height}$, where ‘A’ is the area of the triangle.

From the given figure, we can clearly see that triangle XYZ is a right angle triangle. Therefore, considering ‘XY’ as the base and ‘ZY’ as the height of the triangle, we have,
$\text{A}=\dfrac{1}{2}\times XY\times YZ$
Substituting, XY = 6 m and YZ = 8 m, we get,
$\begin{aligned} & \text{A}=\dfrac{1}{2}\times 6\times 8 \\\ & \Rightarrow \text{A}=\dfrac{48}{2} \\\ & \Rightarrow \text{A}=24\text{ }{{\text{m}}^{2}}..................(i) \\\ \end{aligned}$
Now, we can also find the area of the triangle XYZ by considering ‘XZ’ as the base and ‘YW’ as the height of the triangle. Therefore,
$\text{A}=\dfrac{1}{2}\times ZX\times YW$
Substituting, XZ = 10 m and YW = h, we get,
$\begin{aligned} & \text{A}=\dfrac{1}{2}\times 10\times h \\\ & \Rightarrow \text{A}=5h..................(ii) \\\ \end{aligned}$
Since, equation (i) and equation (ii) represent the area of the same triangle XYZ, therefore, the area given by these two equations must be equal.
Equating the areas given by equation (i) and equation (ii), we get,
$\begin{aligned} & 24=5h \\\ & \Rightarrow h=\dfrac{24}{5} \\\ & \Rightarrow h=4.8\text{ m} \\\ \end{aligned}$
Hence, option (a) is the correct answer.

Note: One may note that we can also find the area of the triangle XYZ by using heron’s formula because all the three sides have been provided to us, but this will be a lengthy process. So, to solve this question in less time and to get rid of calculation, we have applied the formula: $\text{A}=\dfrac{1}{2}\times \text{base}\times \text{height}$. You may note that the areas are equated because we are finding the area of the same triangle two different times. Never use Pythagoras theorem in the triangles XYW or ZYW to solve this problem because we do not know the length of all the sides in these triangles.