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Mathematics · Class 11 Science

Ch 14Properties of Triangles — Class 11 Mathematics, concept-first.

Theorem. In any with sides opposite angles , where is the circumradius of the triangle.

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Cosine Rule

The sine rule needs at least one known angle paired with its opposite side to get started. But two very common situations don't offer that: you might know two sides and the angle between them (SAS), or all three sides (S…

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Chapter contents

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10.1

Sine Rule

Theorem. In any with sides opposite angles , where is the circumradius of the triangle.

10.2

Cosine Rule

Theorem. In any , Equivalently, and cyclically.

10.3

Projection Rule

Theorem. In any ,

10.4

Tangent Rule (Napier's Analogy)

Theorem. In any , and cyclically for the other pairs of angles.

+Exercise 10(a)i15 questions
  1. Q1In $\triangle ABC$, if $b=3$, $c=4$ and $A=60^\circ$, find the side $a$.Free
  2. Q2In $\triangle ABC$, if $a=5$, $b=7$ and $C=60^\circ$, find the side $c$.Free
  3. Q3In $\triangle ABC$, if $b=6$, $c=10$ and $A=120^\circ$, find the side $a$.Free
  4. Q4Show that $(b+c)\cos A+(c+a)\cos B+(a+b)\cos C=a+b+c$.Preview
  5. Q5Show that $\dfrac{a+b}{c}=\dfrac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}}$.Preview
  6. Q6Show that $\dfrac{a-b}{c}=\dfrac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}}$.Preview
  7. Q7Show that $b^2-c^2=a(b\cos C-c\cos B)$.Preview
  8. Q8Prove that $\tan\left(\dfrac{B-C}{2}\right)=\dfrac{b-c}{b+c}\cot\dfrac{A}{2}$, and verify it for the triangle with $a=13,\,b=14,\,c=15$.Preview
  9. Q9The angle of elevation of the top of a tower from a point on the ground is $30^\circ$. On walking $40$ m towards the tower, the angle of ele…Preview
  10. Q10From the top of a building $60$ m high, the angles of depression of the top and the bottom of a tower are observed to be $30^\circ$ and $60^…Preview
  11. Q11A man observes the angle of elevation of the top of a tower to be $45^\circ$. He walks $30$ m nearer to the tower and observes the angle of…Preview
  12. Q12Find the area of the triangle whose sides are $a=13$, $b=14$, $c=15$.Preview
  13. Q13Find the area of $\triangle ABC$ given $b=10$, $c=12$ and $A=60^\circ$.Preview
  14. Q14In $\triangle ABC$, if $a=5$, $B=45^\circ$ and $C=60^\circ$, find the area of the triangle.Preview
  15. Q15In $\triangle ABC$, if $a=7$, $b=8$, $c=9$, find the angle $A$.Preview
10.5

Half-Angle Formulas

Theorem. With , and cyclically for .

10.6

Area of a Triangle

(1) SAS form: . Proof. Drop a perpendicular from to , meeting it at , so (in right triangle ). Taking as the base, . The other two forms follow by choosing a different base.

10.7

Incircle and Inradius

Theorem 1. . Proof. The incentre is equidistant (distance ) from all three sides. Joining to splits into three triangles with heights on bases respectively. So , giving .

10.8

Excircles and Exradii

Every triangle has, besides its incircle, three excircles — each tangent to one side and to the extensions of the other two. The excircle opposite has radius , and similarly opposite .

+Exercise 10(b)i9 questions
  1. Q1Find the inradius $r$ of the triangle whose sides are $a=13$, $b=14$, $c=15$.Free
  2. Q2Find the inradius $r$ of the triangle whose sides are $a=5$, $b=6$, $c=7$.Free
  3. Q3In $\triangle ABC$, if $A=60^\circ$, $b=8$, $c=10$, find the inradius $r$.Free
  4. Q4Find the exradius $r_1$ (opposite to $A$) of the triangle whose sides are $a=13$, $b=14$, $c=15$.Preview
  5. Q5Find the exradius $r_2$ (opposite to $B$) of the triangle whose sides are $a=5$, $b=6$, $c=7$.Preview
  6. Q6Prove that $r_1+r_2+r_3-r=4R$.Preview
  7. Q7Prove that $\dfrac{1}{r_1}+\dfrac{1}{r_2}+\dfrac{1}{r_3}=\dfrac{1}{r}$.Preview
  8. Q8Prove that $r\,r_1\,r_2\,r_3=\Delta^2$.Preview
  9. Q9Prove that $r_1r_2+r_2r_3+r_3r_1=s^2$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 16 questions16 questions
  1. Q1Prove that: $\cot A + \cot B + \cot C = \dfrac{a^2 + b^2 + c^2}{4\Delta}$.Preview
  2. Q2If $P_1, P_2, P_3$ are altitudes drawn from vertices A, B, C to the opposite sides of a triangle respectively, then show that: (i) $\dfrac{1…Preview
  3. Q3In $\triangle ABC$, show that : $b\cos^2\dfrac{C}{2} + c\cos^2\dfrac{B}{2} = s$.Preview
  4. Q4If A, B, C are angles in a triangle, then prove that : $\sin A + \sin B + \sin C = 4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\cos\dfrac{C}{2}$.Preview
  5. Q5If $r : R : r_1 = 2 : 5 : 12$, then prove that the triangle is right angled at A.Preview
  6. Q6In $\triangle ABC$, if $\sin\theta = \dfrac{a}{b+c}$, then show that $\cos\theta = \dfrac{2\sqrt{bc}}{b+c} \cos \dfrac{A}{2}$.Preview
  7. Q7Show that $b \cdot \cos^2 \dfrac{C}{2} + c \cdot \cos^2 \dfrac{B}{2} = S$. (In $\triangle ABC$).Preview
  8. Q8In triangle $ABC$, if $r_1 = 2$, $r_2 = 3$, $r_3 = 6$ and $r = 1$, then prove that $a = 3$, $b = 4$ and $c = 5$.Preview
  9. Q9In $\triangle ABC$, prove that $\dfrac{\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2}}{\cot A + \cot B + \cot C} = \dfrac{(a+b+c)^2}…Preview
  10. Q10In $\triangle ABC$, if $a:b:c = 7:8:9$, then find $\cos A : \cos B : \cos C$Preview
  11. Q11If A, B, C are the angles of a triangle, prove that $\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C$Preview
  12. Q12If $r_1 = 2$, $r_2 = 3$, $r_3 = 6$ and $r = 1$, prove that $a = 3$, $b = 4$ and $c = 5$.Preview
  13. Q13In $\triangle ABC$, prove that $\cot\dfrac{A}{2} + \cot\dfrac{B}{2} + \cot\dfrac{C}{2} = \dfrac{S^2}{\Delta}$.Preview
  14. Q14In $\triangle ABC$, if $a = 13$, $b = 14$, $c = 15$, then show that $R = \dfrac{65}{8}$, $r = 4$, $r_1 = \dfrac{21}{2}$, $r_2 = 12$ and $r_3…Preview
  15. Q15In $\triangle ABC$, prove that $\cot\dfrac{A}{2} + \cot\dfrac{B}{2} + \cot\dfrac{C}{2} = \dfrac{s^2}{\Delta}$.Preview
  16. Q16In $\triangle ABC$, prove that $\left(\dfrac{1}{r} - \dfrac{1}{r_1}\right)\left(\dfrac{1}{r} - \dfrac{1}{r_2}\right)\left(\dfrac{1}{r} - \df…Preview