## 27th SSC CGL level Solution Set, 2nd on topic Mensuration

This is the 27th solution set of 10 practice problem exercise for SSC CGL exam and 2nd on topic Mensuration. Students should complete the corresponding question set in prescribed time first and then only refer to this solution set.

We will repeat here the method of taking a 10 problem test if you have not gone through it already. And if you have not taken the test yet, you can refer to * SSC CGL level Question Set 27, Mensuration 2*, and then after taking the test come back to this solution.

### Method for taking this 10 problem test and get the best results from the test set:

**Before start,**go through the tutorials**Basic concepts on Geometry 1**,**, Basic concepts on Geometry 2**or any other short but good material to refresh your concepts if you so require.**Basic concepts on Geometry 3****Answer the questions**in an undisturbed environment with no interruption, full concentration and alarm set at 15 minutes.**When the time limit of 15 minutes is over,**mark up to which you have answered,**but go on to complete the set.****At the end,**refer to the answers to the questions to mark your score at 15 minutes. For every correct answer add 1 and for every incorrect answer deduct 0.25 (or whatever is the scoring pattern in the coming test). Write your score on top of the answer sheet with date and time.**Identify and analyze**the problems that**you couldn't do**to learn how to solve those problems.**Identify and analyze**the problems that**you solved incorrectly**. Identify the reasons behind the errors. If it is because of**your shortcoming in topic knowledge**improve it by referring to**only that part of concept**from the best source you get hold of. You might google it. If it is because of**your method of answering,**analyze and improve those aspects specifically.**Identify and analyze**the**problems that posed difficulties for you and delayed you**. Analyze and learn how to solve the problems using basic concepts and relevant problem solving strategies and techniques.**Give a gap**before you take a 10 problem practice test again.

Important:bothandpractice testsmust be timed, analyzed, improving actions taken and then repeated. With intelligent method, it is possible to reach highest excellence level in performance.mock tests

**Resources that should be useful for you**

**You may refer to:**

* 7 steps for sure success in SSC CGL tier 1 and tier 2 competitive tests* or

*to access all the valuable student resources that we have created specifically for SSC CGL, but*

**section on SSC CGL****generally for any hard MCQ test.**

**Tutorials that you should refer to**

**If you like, **you may * subscribe* to get latest content from this place.

### 27th solution set - 10 problems for SSC CGL exam: topic Mensuration - Answering time 15 mins

**Problem 1.**

The area of a rectangle is thrice that of a square. The length of the rectangle is 20 cm and the breadth $\frac{3}{2}$ times the length of a side of the square. The side length of the square (in cm) is,

- 10
- 20
- 60
- 30

**Solution 1 - Problem analysis and solving**

By given statement area of the rectangle,

$A_r = 20\times{\text{breadth}} = 20\times{\frac{3}{2}a}$, where $a$ is length of a side of the square,

$=30a$

$=3A_s$, where $A_s$, is the area of the square.

So,

$A_s = 10a = a^2$, by definition.

Or, $a = 10$.

**Answer:** Option a: 10.

**Key concepts used:** Area of rectangle and square concepts.

**Problem 2.**

The perimeter of a rhombus is 40 cm and its height is 5 cm. Its area in (cm$^2$) is,

- 45
- 60
- 55
- 50

**Solution 2 - Problem analysis and solving**

Let us see how a rhombus behaves and what are its basic characteristics. In the figure below ABCD is a rhombus all four sides of which are of equal length $a$,

As length of all sides of a rhombus is equal in length, its side length = $\displaystyle\frac{40}{4} = 10$ cm.

In a rhombus, its area = base side $\times{}$ height = a $\times{}$ h. If you push the rhombus from right to left holding its corner A and keeping its base CD fixed, it adjusts its shape to the rectangle shape of CDGF without losing or adding any area. The area of the triangles $\triangle AGD$ and $\triangle BFC$ are same.

As area of the rectangle = a $\times{}$ h, we get the area of the rhombus as = base $\times{}$ height.

Thus area of the rhombus in our problem = side length $\times{}$ height = 50 cm.

**Answer:** Option d : 50.

**Key concepts used:** Basic concepts on area of rhombus.

**Problem 3.**

The sum of the length, breadth and height of a rectangular parallelepiped is 24 cm and the length of the diagonal is 15 cm. Then its total surface area is,

- 351 cm$^2$
- 256 cm$^2$
- 265 cm$^2$
- 315 cm$^2$

**Solution 3 - Problem analysis and visualization**

The following is the figure corresponding to the problem.

Let the length, breadth and height be, L, B and H. The base diagonal,

$P^2= L^2 + B^2$.

Again, Main diagonal,

$D^2 = P^2 + H^2$

$\hspace{7mm}= L^2 + B^2 + H^2 = 225$.

Given $L + B + H = 24$ cm.

Knowing that the total surface area is, $2(LB +BH+HL)$ using the square of three term sum expression,

$(L +B + H)^2 $

$= L^2 + B^2 + H^2 + 2(LB + BH + HL)$, we get total surface area as,

$2(LB + BH + HL) $

$= 24^2 - 225 $

$= 576 - 225$

$= 351$ cm$^2$.

**Answer:** a: 351 cm$^2$.

**Key concepts used:** Diagonal and area concepts on rectangular parallelepiped.

**Problem 4.**

In the figure given below two shaded regions are formed in a circle of radius $a$.

The area of the shaded region is,

- $\frac{a^2}{2}(\frac{\pi}{2} - 1)$
- $a^2(\pi - 1)$
- $\frac{a^2}{2}(\pi - 1)$
- $a^2(\frac{\pi}{2} - 1)$

#### Solution 4.

The triangle formed by the horizontal diameter is a right triangle (as all diameters subtend a $90^0$ angle at the periphery) with two inclined sides equal and the vertical radius as the perpendicular bisector of the base.

Area of this triangle = $a^2$.

Area of the semi-circle = $\frac{1}{2}\pi{a^2}$.

So, area of the shaded region = $\frac{1}{2}\pi{a^2} - a^2 = a^2(\frac{\pi}{2} - 1)$.

**Answer:** Option d: $a^2(\frac{\pi}{2} - 1)$.

**Key concepts used:** Visualization -- Circle and triangle area concept.

**Problem 5.**

Three circles of radii $a$, $b$ and $c$ touch each other externally. The area of the triangle formed by the three centres is,

- $ab + bc + ca$
- $(a + b +c)\sqrt{ab + bc + ca}$
- $\sqrt{(a + b + c)abc}$
- none of the above

**Solution 5.**

The following figure represents the solution.

The three sides of the triangle are,

$k = a + b$

$l = b + c$, and

$m = c + a$.

The half perimeter of the triangle is then,

$s = \frac{1}{2}(k + l +m) = a + b + c$.

Using the relation of area of triangle with its half perimeter, area

$A = \sqrt{s(s - x)(s-y)(s-z)}$ where $x$, $y$ and $z$ are the three side lengths.

In our case the area of the triangle is then,

$A = \sqrt{(a + b + c)abc}$.

**Answer:** Option c: $\sqrt{(a + b + c)abc}$.

**Key concept used:** Visualization to form length of sides of triangle in terms of radii of the three circles -- using the standard relation between the length of sides of the triangle and its area.

**Problem 6.**

The circumference of a circle is 11 cm. The area of a sector of the circle subtending an angle of $60^0$ at the periphery (take $\pi = \frac{22}{7})$ is,

- $1\displaystyle\frac{29}{48}$ cm$^2$
- $2\displaystyle\frac{27}{48}$ cm$^2$
- $2\displaystyle\frac{29}{48}$ cm$^2$
- $1\displaystyle\frac{27}{48}$ cm$^2$

**Solution 6 **

The visualized figure for the problem is as below.

The circumference is,

$2\pi{r} = 11$, where $r$ is radius

Or, $2\times{22} r = 7\times{11}$,

Or, $r = \displaystyle\frac{7}{4}$ cm

Area subtended by $60^0$ sector is one-sixth of the total area ($60^0$ is one-sixth of $360^0$ the whole angle covering the circle).

Thus area of the sector = $\displaystyle\frac{1}{6}\times{\frac{22}{7}}\times{\frac{49}{16}} = \displaystyle\frac{77}{48} = 1\frac{29}{48}$ cm$^2$.

**Answer:** Option a : $1\displaystyle\frac{29}{48}$ cm$^2$.

**Key concepts used:** Sector and circle area concepts.

**Problem 7.**

If the difference between the areas of the circumcircle and incircle of an equilateral triangle is 44 cm$^2$, then the area of the triangle (in cm$^2$, take $\pi = \frac{22}{7}$), is,

- $28$ cm$^2$
- $21$ cm$^2$
- $7\sqrt{3}$ cm$^2$
- $14\sqrt{3}$ cm$^2$

** Solution 7 **

The corresponding figure is as below.

The circumcentre and incentre of the equilateral triangle is the same point P and the perpendicular bisector CD and the other two such bisectors from vertices A and B intersect at P which in this case is also the centroid and CD is a median.

Let's assume side length of the triangle as $a$.

So,

$CD = \sqrt{a^2 - \left(\displaystyle\frac{1}{2}a\right)^2} = \displaystyle\frac{\sqrt{3}}{2}a$.

$CD$ being the median, $PD$ is one-third of $CD$, that is,

$PD=\displaystyle\frac{1}{2\sqrt{3}}a$

Similarly, $CP$ is two-third of $CD$, that is,

$CP=\displaystyle\frac{1}{\sqrt{3}}a$.

Thus difference in the areas of the circles,

$A_c - A_i = \pi(CP^2 - PD^2) $

$\hspace{18mm}= \pi{a^2}\left(\displaystyle\frac{1}{3} - \displaystyle\frac{1}{12}\right)$

$\hspace{18mm}=\displaystyle\frac{1}{4}\pi{a^2}$

$\hspace{18mm}=44$

Or, $a^2 = 56$.

The area of the equilateral triangle with side length $a$ is,

$A_t = \displaystyle\frac{\sqrt{3}}{4}a^2 $

$\hspace{6mm}= \displaystyle\frac{\sqrt{3}}{4}56$

$\hspace{6mm}=14 \displaystyle\sqrt{3}$ cm$^2$.

**Answer: d: **$14\sqrt{3}$ cm$^2$.

**Key concepts used:** Concepts of circumcentre, incentre, median and areas of circles and equilateral triangles.

**Problem 8.**

If area of an equilateral triangle is $A$ and its height is $b$, the value of $\displaystyle\frac{b^2}{A}$ is,

- $\displaystyle\frac{1}{3}$
- $3$
- $\sqrt{3}$
- $\displaystyle\frac{1}{\sqrt{3}}$

** Solution 8 **

The following figure depicts the problem situation.

If the side length of an equilateral triangle is $a$, its height is,

$b^2 = a^2 - \displaystyle\frac{1}{4}a^2 = \displaystyle\frac{3}{4}a^2$,

And its area is,

$A = \displaystyle\frac{\sqrt{3}}{4}a^2$,

So desired expression,

$\displaystyle\frac{b^2}{A} = \displaystyle\frac{\displaystyle\frac{3}{4}a^2}{\displaystyle\frac{\sqrt{3}}{4}a^2} = \sqrt{3}$.

**Answer:** Option c: $\sqrt{3}$.

**Key concepts used:** Expressing both height $b$ and area $A$ in terms of link variable or intermediate variable side length $a$ simplifies the procedure.

**Problem 9.**

ABCD is a parallelogram. P and Q are the mid-points of sides BC and CD respectively. If the area of the $\triangle ABC$ is 12 cm$^2$, the area of $\triangle APQ$ is,

- 9 cm$^2$
- 12 cm$^2$
- 10 cm$^2$
- 8 cm$^2$

**Solution 9 - Problem analysis **

The following figure corresponds to visualization of the problem.

P being midpoint of BC, it divides the $\triangle ABC$ into two equal parts (height being same, base being half of larger base), that is,

Area of $\triangle APC = \displaystyle\frac{1}{2}$ of area of $\triangle ABC = 6$ cm$^2$.

Similarly, area of $\triangle AQC$ is half of area of $\triangle ACD = 6$ cm$^2$.

So area of quadrilateral $APCQ = 12$ cm$^2$.

Now we only have to find the area of $\triangle PCQ$ and subtract it from this area of the quadrilateral to get the desired area of $\triangle APQ$.

**Solution 9 - Problem solving **

The other diagonal BD also divides the area of the parallelogram into two equal parts and so, area of the $\triangle BCD = 12$ cm$^2$.

Again $BD||PQ$ and P and Q are the midpoints of the other two sides BC and CD of the $\triangle BCD$. Thus these two triangles $\triangle PCQ$ and $\triangle BCD$ are similar and each side including the height of the smaller triangle is half its corresponding side and the height of the larger triangle.

This makes the area of the $\triangle PCQ = \displaystyle\frac{1}{4}$th of the area of $\triangle BCD = 3$ cm$^2$.

Finally then, the area of $\triangle APQ = 12 - 3 = 9$ cm$^2$.

**Answer:** Option a: 9 cm$^2$.

**Key concepts used:** Visualization of relationships between the important areas of triangles -- parallelogram concepts.

** Problem 10.**

A wire when bent in the form of a square encloses an area of 484 sq cm. What will be the enclosed area when the same wire is bent into the form of a circle?

- 616 sq cm
- 693 sq cm
- 462 sq cm
- 539 sq cm

**Solution 10. **

If the side of the square is $a$ cm,

$a^2 = 484 = 22^2$.

So, $a = 22$ cm.

Periphery of the square is then = $4\times{22} = 88$ cm.

This peripheral length of the wire will then enclose a circular area.

So if the radius of the circular area is $r$ cm,

$2\pi{r} = 88$,

Or, $r=14$ cm.

The area of the circular area is then,

$A = \pi{r^2} = \displaystyle\frac{22}{7}\times{14^2} = 616$ sq cm.

**Answer:** a: 616 sq cm.

**Key concepts used:** Transformation of shapes concept.

### Other related guideline, question set and solution set on SSC CGL mensuration

* 7 steps for sure success in SSC CGL tier 1 and tier 2 competitive tests* or

*to access all the valuable student resources that we have created specifically for SSC CGL, but*

**section on SSC CGL****generally for any hard MCQ test.**

**Tutorials that you should refer to**

*Basic concepts on Geometry 1 lines points and triangles*

*Basic concepts on Geometry 2 quadrilaterals polygons squares*

*Basic and rich concepts on Geometry 3 Circles*

*Basic and rich Geometry concepts part 4 Arc angle subtending concept proof*

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