Solution of Triangle

Q.

A statue, $1.6m$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.

Q. Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ},$ respectively, of a triangle $PQR.$

If the triangle $PQR$ varies, then the minimum value of
$\cos (P+Q)+\cos (Q+R)+\cos (R+P)$

1. $-\frac{5}{3}$
2. $\frac{3}{2}$
3. $\frac{5}{3}$
4. $-\frac{3}{2}$

Q. Consider a triangular plot $ABC$ with sides $AB=7m,BC=5m$ and $CA=6m$. A vertical lamp-post at the mid point $D$ of $AC$ subtends an angle ${30}^{\circ }$ at $B$. The height (in $m$) of the lamp-post is :

1. $\frac{3}{2}\sqrt{21}$
2. $7\sqrt{3}$
3. $2\sqrt{21}$
4. $\frac{2}{3}\sqrt{21}$

Q.

What are the formulae for $\cos (c+d)$ and $\cos (c-d)$.

Q.

If the median of $\Delta ABC$ through $A$ is perpendicular to $AB$ then

1. $tanA+tanB=0$

2. None of these

3. $2tanA+tanB=0$

4. $tanA+2tanB=0$

Q. In a $\Delta ABC,$ let $a,b$ and $c$ denote the length of sides opposite to vertices $A,B$ and $C$ respectively. If $b=2,$ $c=\sqrt{3}$ and $\angle BAC=\frac{\pi }{6},$ then value of circumradius of triangle $ABC$ is

Q. The lengths of the sides of a triangle (in units) $ABC$ are $AB=10,BC=7,CA=\sqrt{37}$ then length of the median through vertex $C$ is

1. $3\sqrt{2}$ units
2. $2\sqrt{3}$ units
3. $3\sqrt{3}$ units
4. $4\sqrt{2}$ units

Q.

Given $\cos A=0.25$, find $\angle A$ in degrees. Round the value to the nearest hundredth.

Q. If $AD,BE$ and $CF$ are the medians of $\Delta ABC,$ then the value of $(A{D}^2+B{E}^2+C{F}^2):(B{C}^2+C{A}^2+A{B}^2)$ is:

1. $\frac{3}{4}$
2. $\frac{3}{2}$
3. $\frac{1}{4}$
4. $\frac{1}{2}$

Q.

If the median AD of a triangle ABC is perpendicular to AB then the value of tan A+ 2tan B

1. None of these

2. 0

3. 1

4. 3

Q. If P is a 3×3 orthogonal matrix α, β, γ are the angles made by a straight line OX, OY, OZ and A=⎡⎢⎣sin2αsinαsinβsinαsinγsinαsinβsin2βsinβsinγsinαsinγsinβsinγsin2γ⎤⎥⎦&Q=PTAP.

If PQ6PT=KA then k is

Q. In the given figure, $\Delta ABC$ is a right angled triangle at $A$. The hypotenuse is divided into $n$ equal parts. Let $PQ$ be the segment that contains the midpoint of the hypotenuse.


If $a$ and $h$ be the length of the hypotenuse and altitude to the hypotenuse of the triangle respectively. Then which of the following option is correct ?
(Given that $n$ is an odd integer and $\alpha =\angle PAQ$)

1. $\cot \alpha =\frac{4nh}{(n^2-1)a}$
2. $\tan \alpha =\frac{4an}{(n^2-1)h}$
3. $\cot \alpha =\frac{4an}{(n^2-1)h}$
4. $\tan \alpha =\frac{4nh}{(n^2-1)a}$

Q. Let $AD$ be a median of the $△ABC$. If $AE$ and $AF$ are medians of the triangle $ABD$ and $ADC$, respectively, and $AD=m_1,AE=m_2,AF=m_3$, then $\frac{a^2}{8}$ is equal to

1. $m_1^2+m_3^2-2m_2^2$
2. $m_2^2+m_3^2-2m_1^2$
3. $m_1^2+m_2^2-2m_3^2$
4. $m_1^2+m_2^2-2m_3^2$

Q. Which of the following expressions have value equal to four times the area of the triangle $ABC$?
(All symbols used have their usual meaning in a triangle)

1. $rs+r_1(s-a)+r_2(s-b)+r_3(s-c)$
2. $\frac{(a+b+c{)}^2}{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}$
3. $(a^2+b^2-c^2)\tan B$
4. $b^2\sin 2C+c^2\sin 2B$

Q. In a triangle $ABC,$ points $X$ and $Y$ are on $AB$ and $AC,$ respectively, such that $XY$ is parallel to $BC.$ Which of the following two equalities always hold? (Here $\left[PQR\right]$ denotes the area of triangle $PQR.$)
(I) $\left[BCX\right]=\left[BCY\right]$
(II) $\left[ACX\right]\cdot \left[ABY\right]=\left[AXY\right]\cdot \left[ABC\right]$

1. Neither (I) nor (II)
2. (II) only
3. (I) only
4. Both (I) and (II)

Q.

If AD, BE and CF are the medians of a $△ABCthen(A{D}^2+B{E}^2+C{F}^2):(B{C}^2+C{A}^2+A{B}^2)$ is equal to

1. 4:3

2. 3:2

3. 3:4

4. 2:3

Q. Let $ABC$ be a triangle with $G_1,G_2,G_3$ as the mid-points of $BC\left(a\right),AC\left(b\right)$ and $AB\left(c\right)$ respectively. Also, let $M$ be the centroid of the triangle. It is given that the circumcircle of $\Delta MAC$ touches the side $AB$ of the triangle at point $A$, then

1. $\frac{A{G}_1}{b}=\frac{\sqrt{3}}{2}$
2. $max\{\sin \angle CAM+\sin \angle CBM\}=\frac{2}{\sqrt{3}}$
3. $\frac{a^2+b^2}{c^2}=2$
4. $\frac{c^2}{ab}=\sqrt{2}$ if $\sin \angle CAM+\sin \angle CBM$ is maximum

Q. If b > c sin B, b < c and B is acute angle then number of triangles possible following the given conditions is 1.

1. True
2. False

Q. Let $P$ be a point inside a triangle $ABC$ with $\angle ABC={90}^{\circ }$. Let $P_1$ and $P_2$ be the images of $P$ under reflection in $AB$ and $BC$ respectively. The distance between the circumcenters of triangles $ABC$ and $P_{1}P{P}_2$ is

1. $\frac{AB}{2}$
2. $\frac{AP+BP+CP}{3}$
3. $\frac{AC}{2}$
4. $\frac{AB+BC+AC}{2}$

Q. Which of the following expressions have value equal to four times the area of the triangle $ABC$?
(All symbols used have their usual meaning in a triangle)

1. $rs+r_1(s-a)+r_2(s-b)+r_3(s-c)$
2. $\frac{(a+b+c{)}^2}{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}$
3. $(a^2+b^2-c^2)\tan B$
4. $b^2\sin 2C+c^2\sin 2B$

Q. Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\left(Typeof△ABC\right)\\ \text{a.}\cot \frac{A}{2}=\frac{b+c}{a}& \text{p. always right angled}\\ \text{b.}a\tan A+b\tan B=(a+b)\tan \frac{A+B}{2}& \text{q. always isosceles}\\ \text{c.}a\cos A=b\cos B& \text{r. may be right angled}\\ \text{d.}\cos A=\frac{\sin B}{2\sin C}& \text{s. may be right angled isosceles}\end{array}$

1. $a-p,q;b-q,r;c-p,s;d-q,r$
2. $a-p,q;b-q,s;c-p,s;d-q,s$
3. $a-p,s;b-q,r,s;c-r,s;d-q,r,s$
4. $a-p,s;b-q,r;c-p,s;d-q,r$

Q. Let ABC be a triangle with incentre I and r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB respectively, If $r_1,r_2,and{r}_3$ are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that
$\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1{r}_2{r}_3}{(r-r_1)(r-r_2)(r-r_3)}$

1. True
2. False

Q. The top of a ladder reaches a point on the wall $5m$ above the ground. If the foot of the ladder makes an angle of ${30}^{\circ }$ with the ground, find the length of the ladder.

Q. In $\Delta$ ABC having vertices $A(acos{\theta }_1,asin{\theta }_1),B(acos{\theta }_2,asin{\theta }_2)$ and $C(acos{\theta }_3,asin{\theta }_3)$ is equilateral, then which of the followings is/are true?

1. $cos{\theta }_1+cos{\theta }_2+cos{\theta }_3=0$
2. $sin{\theta }_1+sin{\theta }_2+sin{\theta }_3=0$
3. $cos({\theta }_1-{\theta }_2)+cos({\theta }_2-{\theta }_3)+cos({\theta }_3-{\theta }_1)=\frac{-3}{2}$
4. $\left|sin\left(\frac{{\theta }_1-{\theta }_2}{2}\right)\right|=sin\left|\left(\frac{{\theta }_2-{\theta }_3}{2}\right)\right|=\left|sin\left(\frac{{\theta }_1-{\theta }_3}{2}\right)\right|$

Q. In a triangle ABC, $r_1,r_2,r_3$ are the ex-radii of the ex-circles. If $a=\frac{r_1+r_2+r_3}{r}\text{and}b=\frac{r_1{r}_2{r}_3}{r^3},$ where r is the inradius then the minimum value of $(\left(\frac{a_{min}}{3}\right){\tan }^{2}A+\left(\frac{b_{min}}{9}\right){\cot }^{2}A)$ is

Q. Let $ABC$ be a triangle with $G_1,G_2,G_3$ as the mid-points of $BC\left(a\right),AC\left(b\right)$ and $AB\left(c\right)$ respectively. Also, let $M$ be the centroid of the triangle. It is given that the circumcircle of $\Delta MAC$ touches side $AB$ of the triangle at point $A$, then

1. $\frac{A{G}_1}{b}=\frac{\sqrt{3}}{2}$
2. $max\{\sin \angle CAM+\sin \angle CBM\}=\frac{2}{\sqrt{3}}$
3. $\frac{a^2+b^2}{c^2}=2$
4. $\frac{c^2}{ab}=\sqrt{2}$ if $\sin \angle CAM+\sin \angle CBM$ is maximum

Q. In a right angled triangle, altitude divides the hypotenuse into two segments of length $8$ units and $18$ units, then the length of the smallest side is
(correct answer + 1, wrong answer - 0.25)

1. $8\sqrt{2}\text{units}$
2. $6\sqrt{13}\text{units}$
3. $4\sqrt{13}\text{units}$
4. $8\sqrt{3}\text{units}$

Q.

Given angle $A={60}^{\circ },c=\sqrt{3}-1,b=\sqrt{3}+1$. Solve the triangle

1. $a=\sqrt{6},B={15}^{\circ },C={100}^{\circ }$
2. $a=\sqrt{12},B={15}^{\circ },C={105}^{\circ }$
3. $a=\sqrt{6},B={105}^{\circ },C={15}^{\circ }$
4. $a=\sqrt{6},B={15}^{\circ },C={105}^{\circ }$

Q. Given angle $A={60}^{\circ },c=\sqrt{3}-1,b=\sqrt{3}+1$. Solve the triangle

1. $a=\sqrt{6},B={15}^{\circ },C={100}^{\circ }$
2. $a=\sqrt{6},B={15}^{\circ },C={105}^{\circ }$
3. $a=\sqrt{12},B={15}^{\circ },C={105}^{\circ }$
4. $a=\sqrt{6},B={105}^{\circ },C={15}^{\circ }$

Q. Number of triangles possible for a given b, c and B(acute angle) under the condition that b < c sin B. Where b, c are the sides and B is the angle opposite to b.

1. 2
2. 3
3. 0
4. 1