Question

The figure above is the floor plan drawn by an architect for a small concert hall. The stage has depth 8 meters (m) and two walls each of

length 10 m. If the seating portion of the hall has an area of 180 square meters, what is the value of *x *?

Hint:

**Hint:**

We need to know Pythagoras theorem and the formula of area of a rectangle to solve this question. Pythagoras theorem states that, in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Also, the seating area is in the shape of a rectangle and area of a rectangle is given by the product of its length and breadth.

## The correct answer is: 15

### First, we find the breadth of the seating are, which is equal to length of the base of the stage.

ABC is an isosceles triangle, so the perpendicular from the vertex on the base is its mid-point.

Let F be the midpoint of BC.

Clearly, AFC is a right angled triangle with *angle AFC = *90°.

So, by Pythagoras theorem, we have

Given,

Using these values in the above equation, we have

Simplifying, we get

Hence, value of FC is given by

As F is the mid-point of BC, so

Thus, the length of the base of the triangle ABC is given as

Thus,

Breadth of the sitting area = 12 m

Length of the sitting area = x m

Area of the sitting area = 180 m^{2}

We know,

So, we have

Thus, the value of x is given by

Thus, the correct answer is 15.

**Note:**

There are different concepts used to solve this problem. Another concept used which is not mentioned in the hint is that the perpendicular drawn from the vertex of an isosceles triangle to the

base cuts the base in half. This property is also applicable in equilateral triangles as they are a special case of isosceles triangle.

### Related Questions to study

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### In the figure, ABCD is a rectangle and PQRS is a square. Find the area of the shaded portion.

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The scatterplot above shows the revenue, in millions of dollars, that a company carned over several years and a line of best fit for the data. In Year 4 , the difference between the actual revenue and the predicted revenue is n million dollars, where is a posative integer. What is the value of n ? Round your answer to the nearest whole number. (Disregard the 5 sign when gridding your answer.)

**Note:**

As we are not certain if the point of the actual revenue is exactly at the midpoint of 50 and 60, so instead of taking the midpoint 55, we can also take it to be 54 or 56.

Then the value of *n* can be 4 or 6 too.

The scatterplot above shows the revenue, in millions of dollars, that a company carned over several years and a line of best fit for the data. In Year 4 , the difference between the actual revenue and the predicted revenue is n million dollars, where is a posative integer. What is the value of n ? Round your answer to the nearest whole number. (Disregard the 5 sign when gridding your answer.)

**Note:**

As we are not certain if the point of the actual revenue is exactly at the midpoint of 50 and 60, so instead of taking the midpoint 55, we can also take it to be 54 or 56.

Then the value of *n* can be 4 or 6 too.