# S.L. Loney Quadrilaterals and Regular Polygons Solutions (Chapter 16)

SL Loney Plane Trigonometry Solutions for Chapter 16 ‘Quadrilaterals and Regular Polygons’ is an important guide to prepare for IIT JEE, NEET and other tough entrance examinations. This solution will help you cement all the concepts of the quadrilaterals and regular polygons. You will also learn how to determine the area of a pentagon, a hexagon, an octagon, decagon and a dodecagon. With the help of our solutions and the right amount of practice, you can achieve a complete understanding of the chapter.

SL Loney Solutions for Quadrilaterals and Regular Polygons includes 2 exercises with 32 questions. By practising these questions, you will get the knowledge about finding areas of a quadrilateral inscribed in a circle, area of quadrilateral not inscribed in a circle, the radius of the incircle and circumcircle. Not only this, but there are also topics like the area of a polygon, numbers of sides of the polygon, the tangent of an angle which you will master.

Mastering any part of geometry requires a high level of dedication and appropriate material. All the topics are elaborated in a possible simple manner in InstaSolv Solutions to make the reasoning very thorough. Not only the concepts are covered efficiently but it also includes illustrations which are solved in detail to get the peak level of clarity.

## Important Topics for SL Loney Plane Trigonometry Solutions Chapter 16: Quadrilaterals and Regular Polygons

**Area of a Quadrilateral**

A quadrilateral is a closed figure on one plane which has four sides, four angles and four vertices.

Instances of quadrilaterals are rectangle, trapezium, rhombus, parallelogram and square.

To determine the area of a quadrilateral, we divide it into two geometric figures. Let’s say a triangle. Then we find the area of these two triangles individually and at the end add both of the triangle areas. This way we obtain an area of a quadrilateral.

- Quadrilateral not inscribed in a circle

We can evaluate the area of a quadrilateral which is not particularly inscribed in a circle by calculating its area in terms of the sides and by adding any two opposite angles. When the lengths of the quadrilateral are is *Δ *known, the area of *Δ *is the highest.

Here, = area of ABC+area of ACD

Thus, we get *a^{2} + b^{2} – c^{2} -d^{2 }*= 2ab cos B – 2cd cos D

Squaring both the sides of the equation we obtain,

*16 **Δ ^{2 }*

*= (ab + cd)*

*– (a*^{2}*+ b*^{2}*– c*^{2}*-d*^{2}*)*^{2}*– 16abcd cos*^{2}*a*^{2}Finally the area of Quadrilateral becomes Δ^{2}= *(s-a)(s-b)(s-c)(s-d)-abcd cos ^{2} a^{ }*

**A quadrilateral with the Circle Inscribed**

Let’s say we have a quadrilateral MNOP and its sides MN, NO, OP and PM touch the circle in the points E,F,G and H.

So, we have ME=MH, NF=NG, OF=OG and PG=PH

That is, MN+OP=NO+PM

Again, it is, m+o = n+p

Hence, *s= *

Therefore, the area required =

**Regular Polygon**

A regular polygon is both equiangular and equilateral. It means all the angles and sides are of equal measures.

**Area of a regular polygon**

### Discussion of Exercises of SL Loney Plane Trigonometry Solutions Chapter 16: Quadrilaterals and Regular Polygons

- In the first part of the SL Loney Plane Trigonometry Solutions, there are 12 questions. Initially, you will witness the questions related to the area of quadrilaterals. But as you move further, most of the questions are about proving the given statements.
- The second part of the chapter involves 20 questions. Here you will notice questions where it is asked to find the tangent of an angle, number of sides of polygons, value of
*n*for the polygon, radius of the circumcircle. Moreover, there are a few questions about where you will be asked to prove the given equations. - In the third question under the second part of chapter 16, there are multiple questions where you will have to find the area of various quadrilaterals. This can be determined with the help of formulas. As the quadrilaterals are different so are the formulas for each of them.

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