Chapter 13

In any triangle, if \(\tan \frac{A}{2}=\frac{5}{6}\) and \(\tan \frac{B}{2}=\frac{20}{37}\), find \(\tan \frac{C}{2}\) and prove that in this triangle \(a+c=2 b\).

In a \(\Delta A B C, a=13, b=14, c=15\), then find \(\sin \frac{A}{2}\).

If \(a, b\) and \(c\) be in A.P. prove that i. \(\cot \frac{A}{2}, \cot \frac{B}{2}\) and \(\cot \frac{C}{2}\) are in A.P. ii. \(\cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}\) and \(\cos C \cot \frac{C}{2}\) are in A.P. iii. \(a \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{A}{2}=\frac{3 b}{2}\). iv. \(\tan \frac{A}{2}+\tan \frac{C}{2}=\frac{2}{3} \cot \frac{B}{2}\). v. \(\cot \frac{A}{2} \cot \frac{C}{2}=3\).

If \(a, b\) and \(c\) are in H.P., prove that \(\sin ^{2} \frac{A}{2}, \sin ^{2} \frac{B}{2}\) and \(\sin ^{2} \frac{C}{2}\) are also in H.P.

The sides of a triangle are in A.P. and the greatest and least angles are \(\theta\) and \(\phi\); prove that \(4(1-\cos \theta)(1-\cos \phi)=\cos \theta+\cos \phi\)

The sides of a triangle are in A.P. and the greatest angle exceeds the least by \(90^{\circ}\); prove that the sides are proportional to \(\sqrt{7}+1, \sqrt{7}\) and \(\sqrt{7}-1\).

If \(C=60^{\circ}\), then prove that \(\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}\).

In any triangle prove that, if \(\theta\) be any angle, then \(b \cos \theta=c \cos (A-\theta)+a \cos (C+\theta)\).

If \((a+b+c)(b+c-a)=k b c\), then prove that \(k \in(0,4)\).

The sides of a triangle are \(a, b, \sqrt{a^{2}+a b+b^{2}}\), prove that the greatest angle is \(120^{\circ} .\)

In any triangle \(A B C\), if \(\sin ^{2} A+\sin ^{2} B=\sin ^{2} C\), then show that the triangle is right angled.

In any \(\triangle A B C\) if \(2 \cos B=\frac{a}{c}\), then show that the triangle is isosceles.

If in a \(\triangle A B C, a \sin A=b \sin B\), then show that the triangle is isosceles.

If \(\frac{\cos A+2 \cos C}{\cos A+2 \cos B}=\frac{\sin B}{\sin C}\), prove that the triangle is either isosceles or right angled.

If \(\frac{2 \cos A}{a}+\frac{\cos B}{b}+\frac{2 \cos C}{c}=\frac{a}{b c}+\frac{b}{c a}\), find the value of \(A .\left\\{\right.\) Ans. \(\left.90^{\circ}\right\\}\)

If the angles of a triangle are in the ratio \(1: 2: 4\), then prove that \(a^{2} b^{2} c^{2}=\left(b^{2}-a^{2}\right)\left(c^{2}-b^{2}\right)\left(c^{2}-a^{2}\right)\).

\(\frac{a^{2}-b^{2}}{a^{2}+b^{2}}=\frac{\sin (A-B)}{\sin (A+B)}\), prove that the triangle is either isosceles or right angled.

If \(\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1\), prove that the triangle is right angled.

If \(\cot A+\cot B+\cot C=\sqrt{3}\), prove that the triangle is equilateral.

If \(b+c=3 a\), prove that \(\cot \frac{B}{2} \cot \frac{C}{2}=2\).

In a triangle \(A B C, \angle B=\frac{\pi}{3}\) and \(\angle C=\frac{\pi}{4} .\) Let \(D\) divide \(B C\) internally in the ratio \(1: 3\), then find the value of \(\frac{\sin \angle B A D}{\sin \angle C A D}\).

The sides of a triangle are three consecutive natural numbers and it's largest angle is twice the smallest one. Determine the sides of the triangle.

Find the greatest angle of the triangle whose sides are \(x^{2}+x+1,2 x+1\) and \(x^{2}-1\).

If \(c(a+b) \cos \frac{B}{2}=b(a+c) \cos \frac{C}{2}\), prove that the triangle is isosceles.

If \(B=3 C\), prove that \(\cos C=\sqrt{\frac{b+c}{4 c}}, \sin C=\sqrt{\frac{3 c-b}{4 c}}\) and \(\sin \frac{A}{2}=\frac{b-c}{2 c}\).

If \(a, b, c\) be \(5,4,3\) respectively and \(D\) and \(E\) are the points of trisection of side \(B C\), then prove that \(\tan \angle C A E=\frac{3}{8}\).

\(A B C D\) is a trapezium such that \(A B\) is parallel to \(C D\) and \(C B\) is perpendicular to them. If \(\angle A D B=\theta, B C=\) \(p\) and \(C D=q\), show that \(A B=\frac{\left(p^{2}+q^{2}\right) \sin \theta}{p \cos \theta+q \sin \theta}\)

If the tangents of the angles of a triangle are in A.P., prove that the squares of the sides are in the ratio \(x^{2}\left(x^{2}+9\right):\left(3+x^{2}\right)^{2}: 9\left(1+x^{2}\right)\), where \(x\) is the tangent of the least or greatest angle.

If \(p\) and \(q\) be perpendiculars from the angular points \(A\) and \(B\) on any line passing through the vertex \(C\) of the triangle \(A B C\), then prove that \(a^{2} p^{2}+b^{2} q^{2}-2 a b p q \cos C=a^{2} b^{2} \sin ^{2} C\).

In the triangle \(A B C\), lines \(O A, O B\) and \(O C\) are drawn so that the angles \(O A B, O B C\) and \(O C A\) are each equal to \(\omega\); prove that \(\cot \omega=\cot A+\cot B+\cot C\) and \(\operatorname{cosec}^{2} \omega=\operatorname{cosec}^{2} A+\operatorname{cosec}^{2} B+\operatorname{cosec}^{2} C\)

In any triangle \(A B C\) if \(D\) be any point of the base \(B C\), such that \(B D: D C=m: n\), and if \(\angle B A D=\alpha, \angle D A C\) \(=\beta\), and \(\angle C D A=\theta\) and \(A D=x\), prove that \((m+n) \cot \theta=m \cot \alpha-n \cot \beta=n \cot B-m \cot C\) and \((m+n)^{2} \cdot x^{2}=(m+n)\left(m b^{2}+n c^{2}\right)-m n a^{2}\)

Two straight roads intersect at an angle of \(60^{\circ} .\) A bus on one road is \(2 \mathrm{~km}\). away from the intersection and a car on the other is \(3 \mathrm{~km}\). away from the intersection. Find the direct distance between the two vehicles

A ring, \(10 \mathrm{~cm}\). in diameter, is suspended from a point \(12 \mathrm{~cm}\). above its center by 6 equal strings attached to its circumference at equal intervals. Find the cosine of the angle between consecutive strings.

The side of a base of a square pyramid is \(a\) meters and it's vertex is at a height of \(h\) meters above the center of the base. If \(\theta \& \phi\) be respectively the inclinations of any face to the base and of any two faces to one another, prove that \(\tan \theta=\frac{2 h}{a}\) and \(\cot \frac{\phi}{2}=\sqrt{1+\frac{a^{2}}{2 h^{2}}}\).

Find the area of the triangle having sides 13,14 and \(15 \mathrm{~cm}\).

If the angles of a triangle are \(30^{\circ}\) and \(45^{\circ}\) and the included side is \((\sqrt{3}+1) \mathrm{cm} .\), then prove that the area of the triangle is \(\frac{1}{2}(\sqrt{3}+1)\) sq. \(\mathrm{cm}\).

If \(B=45^{\circ}, a=2(\sqrt{3}+1)\) units and \(\Delta=6+2 \sqrt{3}\) sq. units. Determine the side \(b\).

If one angle of a triangle be \(60^{\circ}\), the area \(10 \sqrt{3}\) sq. \(\mathrm{cm}\). and the perimeter \(20 \mathrm{~cm} .\), find the length of the sides.

If \(p_{1}, p_{2}, p_{3}\) are altitudes of a triangle \(A B C\) from the vertices \(A, B, C\) and \(\Delta\) the area of the triangle, then prove that \(p_{1}^{-2}+p_{2}^{-2}+p_{3}^{-2}=\frac{a^{2}+b^{2}+c^{2}}{4 \Delta^{2}}\).

If \(\frac{a}{1+m^{2} n^{2}}=\frac{b}{m^{2}+n^{2}}=\frac{c}{\left(1-m^{2}\right)\left(1+n^{2}\right)}\), then prove that \(\tan \frac{A}{2}=\frac{m}{n}, \tan \frac{B}{2}=m n\) and \(\Delta=\frac{m n b c}{m^{2}+n^{2}}\).

The sides \(a, b, c\) of a triangle are the roots of the equation \(x^{3}-p x^{2}+q x-r=0\). Prove that \(\Delta=\frac{1}{4} \sqrt{p\left(4 p q-p^{3}-8 r\right)} .\)

Through the angular points of a triangle are drawn straight lines which make the same angle \(\alpha\) with the opposite sides of the triangle. Prove that the area of the triangle formed by them is to the area of the original triangle as \(4 \cos ^{2} \alpha: 1\).

In the sides \(B C, C A\) and \(A B\) are taken three points \(A^{\prime}, B^{\prime}, C^{\prime}\) such that \(B A^{\prime}: A^{\prime} C=C B^{\prime}: B^{\prime} A=A C^{\prime}: C^{\prime} B=\) \(m: n\). Prove that if \(A A^{\prime}, B B^{\prime}\) and \(C C^{\prime}\) be joined, they will form by their intersections a triangle whose area is to that of the triangle \(A B C\) as \((m-n)^{2}: m^{2}+m n+n^{2}\)

In a triangle \(A B C, a=4, b=3, \angle A=60^{\circ}\). Then show that \(c\) is the root of the equation \(c^{2}-3 c-7=0\).

In the ambiguous case, given \(a, b, A\) and \(c_{1}, c_{2}\) are the two value of side \(c\), then show that i. \(\quad c_{1}+c_{2}=2 b \cos A\). ii. \(\quad c_{1} c_{2}=b^{2}-a^{2}\). iii. \(c_{1} \sim c_{2}=2 \sqrt{a^{2}-b^{2} \sin ^{2} A}\). iv. \(c_{1}^{2}-2 c_{1} c_{2} \cos 2 A+c_{2}^{2}=4 a^{2} \cos ^{2} A\). v. \(\left(c_{1}-c_{2}\right)^{2}+\left(c_{1}+c_{2}\right)^{2} \tan ^{2} A=4 a^{2}\). vi. \(\frac{(a+b)^{2}}{1+\cos C_{1}}+\frac{(b-a)^{2}}{1-\cos C_{1}}=\frac{(a+b)^{2}}{1+\cos C_{2}}+\frac{(b-a)^{2}}{1-\cos C_{2}}\). vii. \(\cos B_{1} C B_{2}=\frac{2 c_{1} c_{2}}{c_{1}^{2}+c_{2}^{2}}\) if \(A=45^{\circ}\).

In the ambiguous case, given \(a, c, A\) and \(b_{2}=2 b_{1}\), where \(b_{1}, b_{2}\) are the two value of side \(b\), then prove that \(3 a=c \sqrt{1+8 \sin ^{2} A}\)

In the ambiguous case, given \(a, b, A\), if the remaining angles of the triangles formed be \(B_{1}, C_{1}\) and \(B_{2}\), \(C_{2}\), then prove that \(\frac{\sin C_{1}}{\sin B_{1}}+\frac{\sin C_{2}}{\sin B_{2}}=2 \cos A\).

In the ambiguous case, given \(a, b, A\), prove that the sum of the areas of the two triangles formed is \(\frac{1}{2} b^{2} \sin 2\)

If \(2 b=(m+1) a\) and \(\cos A=\frac{1}{2} \sqrt{\frac{(m-1)(m+3)}{m}}\), where \(1

Determine the lengths of medians in terms of the sides.