Tag: similarity and right angled triangle

Questions Related to similarity and right angled triangle

Find the side of the square whose diagonal is $16 \sqrt 2$ cm.

  1. $4$ cm

  2. $16$ cm

  3. $8$ cm

  4. $16\sqrt 2$ cm


Correct Option: B
Explanation:

We know that,

1) All angles of a square are congruent. i.e $90^o$
2) Diagonal of a square bisects each of its angles.
Therefore, the square gets divided into $2$ triangles of degrees $45^o-45^o-90^o$
$\therefore \sin 45^o = \cfrac {\text {side}}{\text {hyp}}$ 
$\therefore \cfrac {1}{\sqrt 2} = \cfrac {\text {side}}{16 \sqrt 2}$
$\therefore$ side of the square $= 16$ cm.

The sides of triangle are in A.P. and the greatest angle exceeds the least by 90. The sides are in the ratio _____________.

  1. $1 : 2 : \sqrt { 2 }$

  2. $1 : \sqrt { 3 } : 2$

  3. $\sqrt { 7 } + 1 : \sqrt { 7 } : \sqrt { 7 } - 1$

  4. $\sqrt { 3 } + 1 : 1 : \sqrt { 3 } - 1$


Correct Option: C

If  H is orthocenter of triangle PQR then PH + QH + RH is 

  1. QR cot P + PR cot Q + PQ cot R

  2. (pq + QR + RP) (cot P + cot Q + copt R)

  3. $\dfrac{1}{2r}(cot P + cotQ + cot R)$

  4. none of these


Correct Option: A

ABC is a triangle right angle at B. D is a point on AC such that $\angle ABD = 45^0$. If AC =$6$ and AD =$2$ , then AB is 

  1. $\dfrac{6}{\sqrt{5}}$

  2. ${3}{\sqrt{2}}$

  3. $\dfrac{12}{\sqrt{5}}$

  4. $2$


Correct Option: A

consider a triangle PQR in which the relation $ QR^2+PR^2=5*PQ^2$ holds. let G be the point of intersection of the medians PM and QN . then angle QGM is always

  1. less then 45 degree

  2. obtuse

  3. a right angle

  4. acute and larger than 45 degree


Correct Option: A

If in a $\Delta ABC,\sin A=\sin^{2} B$ and $2\cos^{2}A=3\cos^{2}B$, then the $\Delta ABC$ is 

  1. Right angled

  2. Obtuse angled

  3. Isosceles

  4. Equilateral


Correct Option: A

Which of the following  can be the sides of a right-angled triangle?

  1. $0.5cm, 1.2 cm, 1.3cm$

  2. $2.4cm, 3.2 cm, 7.9cm$

  3. $5.0cm, 5.25 cm, 7.25cm$

  4. $1.6cm, 3.0 cm, 3.4cm$


Correct Option: A

Let $\Delta _1$ denotes the area of the triangle formed by the vertices $(a^3m^3 _1, am _1), (a^3m^3 _2am _2), (a^3m^3 _3, am _3)$ and $\Delta _2$ denotes the area of the triangle formed by the vertices $(2am _1m _2, a^2(m^2 _1+m^2 _2))$, $(2am _2m _3, a^2(m^2 _2+m^2 _3))$ and $(2am _3m _1, a^2(m^2 _3+m^2 _1))$. Then $\dfrac{\Delta _1}{\Delta _2}(a > 0)$ equals?

  1. $\dfrac{a}{2}$

  2. $2a$

  3. $\dfrac{a^3}{8}$

  4. $8a^3$


Correct Option: A

The sides of $\Delta ABC$ are 5, 7, 8 units then $AG^2 + BG^2 + CG^2$ where G is centroid of $\Delta ABC$ is 

  1. $48.2$

  2. $49.2$

  3. $92.2$

  4. $69.2$


Correct Option: A

For a right-angled triangle, two small sides are of $6$cm and $8$cm length. Length of third side will be

  1. $14$cm

  2. $9$cm

  3. $10$cm

  4. none of these


Correct Option: A