# S.L. Loney Properties of Triangle Solutions (Chapter 15)

SL Loney Plane Trigonometry Solutions for Chapter 15 ‘Properties of a Triangle’ will acquaint you with the fundamentals of how trigonometry and area of a triangle can be combined. With an elaborate description of each topic, explanation of the concepts, and 100% accurate answers, our solutions for SL Loney Trigonometry Solutions Properties of a Triangle is the best guide for you. These solutions will immensely help you in preparing for Class 12 Maths along with JEE and NEET preparations.

SL Loney Plane Trigonometry Solutions for JEE for Properties of a Triangle’ contains 79 questions spread across 3 sections. Every concept is followed by a few questions on that topic. In these questions, you would get to work on various ideas and formulas related to the area of a triangle in terms of trigonometric ratio, learn the different kinds of circles connected with a triangle like a circumcircle/incircle/escribed circle, determine radii of these circles in terms of different sides, what is orthocenter and pedal triangle of a triangle, centroid, and median of a triangle, bisectors of angles.

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## Important Topics for SL Loney Plane Trigonometry Solutions Chapter 15: Properties of a Triangle

**Area of a Triangle – **This shows how to express the area of a triangle as a trigonometric function. In a triangle ABC, if a perpendicular AD is drawn as shown below and a parallel line EAF to BC and drop 2 perpendiculars BE and CF onto this line, then from the diagram we get:

Area of triangle ABC = ½ are of rectangle BF = ½ * BC * CF = ½ * a * CF

From the properties of a right angle triangle we get:

AD = AB Sin B = c Sin B;

Hence the above area, after replacing for AD, can be expressed as:

D = ½ * ca * Sin B = ½ * ab * Sin C = ½ * bc * sin A

We also know from earlier chapters that:

A = 2/bc √(s (s – a) (s – b) (s – c)); Here s – perimeter of triangle/2

Now replacing this in area formula we get:

D = ½ * bc * sin A = √(s (s – a) (s – b) (s – c))

**Circumcircle – **Circle which passes through the angular points A, B, and C of a triangle is called its circumcircle. Its radius is denoted by R

R = a/ (2 Sin A) = b/ (2 Sin B) = c/ (2 Sin C) = abc/4S, S is the area of the triangle.

**Incircle – **If a circle is drawn inside a triangle so that it touches all the sides, it is called its incircle. Its radius is denoted by r

r = S/s

S is Area of the triangle, s is the perimeter of the triangle/2

Another formula is r = (s-a) tan A/2 = (s-b) tan B/2 = (s-c)tan C/2

**Escribed circle – **This exists relative to each angle, so if we draw a circle touching sides BC and the 2 sides AB and AC produced, then it is the escribed circle opposite to angle A. Its radius is denoted by r_{1}

r_{1} = S/s-a, this is for the escribed circle opposite to angle A

r_{2} = S/s-b, this is for the escribed circle opposite to angle B

r_{3} = S/s-c, this is for the escribed circle opposite to angle C

**Orthocenter and Pedal Triangle of any triangle – **In any triangle ABC, if we drop a perpendicular from every angle on its opposite side, then those perpendiculars would meet at a point, let us say P, which is known as the orthocenter of the triangle, and if we draw a triangle by joining the feet of these perpendiculars then we get the pedal triangle of that triangle. Triangle LKM is the pedal angle of the triangle ABC in the figure given below.

**The distance of orthocenter from angular points: **From figure 1.2, the distances PA, PB, and PC are given by 2R * Cos A, 2R * Cos B, and 2R * Cos C, respectively. Here R = a/Sin A = b/sin B = c/Sin C

**Sides and angles of the pedal triangle –** In figure 1.2, KLM is the pedal triangle. Its sides are given by:

**KL = c Cos C, LM = a Cos A, MK = b Cos B**

Angles of a pedal triangle are given by twice the supplements of the angles of the main triangle

**Excentric Triangle** – If we draw lines I_{1}, I_{2}, and I_{3} which touch the points A, B, and C externally and meet, then the triangle they form I_{1}I_{2}I_{3} is called the excentric triangle of any triangle.

**Centroid and Medians of any triangle – **If we take middle points of the sides BC, AC, and AB as d, e, and f respectively then AD, BE, and CF are called medians of that triangle. Euclid showed that medians meet at a common point inside the triangle (let us say G), called the centroid of the triangle, and the following formula can be applied to it:

AG = 2/3 * AD, BG = 2/3 * BE, CG = 2/3 * CF

**Length of the medians – Lengths of the medians can be expressed as:**

AD = ½ * √ (2b^{2} + 2c^{2 }– a^{2}) = ½ * √(b^{2} + c^{2} + 2bc * Cos A)

BE = ½ * √ (2c^{2} + 2a^{2 }– b^{2})

CF = ½ * √ (2a^{2} + 2b^{2 }– c^{2})

If you join circumcenter to the orthocenter, then centroid lies on that line joining the 2 centres.

Distance between circumcenter and orthocenter is given by:

D = R * √ (1 – 8 Cos A 8 Cos B * Cos C)

### Discussion of Exercises of SL Loney Plane Trigonometry Solutions Chapter 15: Properties of a Triangle

- The first part of the exercise has 18 questions in which you need to apply the formula for the area of a triangle to solve various kinds of sums.
- The 2
^{nd}part is dedicated to problems on the circumcircle, incircle, and escribed circles. It has 22 questions where you will be given sides of a triangle and have to find the different radii pertaining to these circles. - The 3
^{rd}part has questions related to the orthocenter and centroid of a triangle and has 39 questions. They are a mix of concepts from on circumcircle, incircle, and escribed circles and their relation with orthocenter and centroid of a triangle, apply theorems to find the area of a triangle formed by these circles.

## Why Use SL Loney Plane Trigonometry Solutions Chapter 15: Properties of a Triangle by Instasolv?

- This is an interesting topic with many theorems on a triangle, circles, and their relations.
- Solving these questions would give you ample practice in various ways this topic can be presented to you in JEE, Mathematics Olympiad kind of competitions. So to ace them you need accurate and simple solutions to Plane trigonometry by SL Loney Solutions.
- Our team has taken efforts to solve each and every problem such that it also clears your concepts and teaches you how to go about a problem in a way that is less time consuming to solve.
- We provide SL Loney Solutions free of cost in a readily accessible online portal which is available for quick revision and learning some tips to be thoroughly prepared for any kind of exam.