Equal Angles Subtend Equal Sides

Q.

If $x+\frac{1}{x}=5$, find the value of $x^2+\frac{1}{x^2}$.

Q.

Find the value of cos $(2\mathrm{\pi }/3)$

Q.

$\frac{\sin \theta +\sin 2\theta }{1+\cos \theta +\cos 2\theta }=$

1. $\frac{1}{2}\tan \theta$

2. $\frac{1}{2}\cot \theta$

3. $\tan \theta$

4. $\cot \theta$

Q.

How to find the value of $\sin \left({300}^{°}\right)$?

Q.

Two vertices of a triangle are $(5,-1)$and $(-2,3).$ If orthocentre is the origin then coordinates of the third vertex are

1. $\left(7,4\right)$

2. $\left(-4,7\right)$

3. $\left(4,-7\right)$

4. $\left(-4,-7\right)$

Q.

The line segment joining the mid - points of any two sides of a triangle is parallel to the third side and equal to half of it.
In the given figure, the side AC of $\Delta ABC$ is produced of E such that $CE=\frac{1}{2}AC$. If D is the mid - points of BC and ED produced meets AB at F and CP, DQ are drawn parallel to BA, then FD =

1. $\frac{1}{2}FE$

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

3. $FE$

4. $\frac{1}{4}FE$

Q.

Which of the following is not true for the figure below?

1. AC ⊥ BD

2. ∠ADB = ∠ ABD

3. ∠DAB = ∠ DCB

4. BC = CD

Q.

In a triangle, the perpendicular from vertex to the base bisects the base. The triangle is

1. Isosceles Triangle

2. Right angled Triangle

3. Equilateral Triangle

4. Scalene Triangle

Q.

Angles opposite the equal sides of a triangle are equal. True or False

1. True

2. False

Q.

Choose the correct option and justify your choice : $\frac{1-{\tan }^{2}45°}{1+{\tan }^{2}45°}=?$

1. $\tan 90°$

2. $1$

3. $\sin 45°$

4. $0$

Q.

In Fig. if $x+y=w+z$, then prove that $AOB$ is a line.

Q.

Prove that $\sin 40°-\cos 70°=\sqrt{3}\cos 80°$

Q.

How do you work out the length of a triangle$?$

Q.

Find out the value of $sec{15}^{\circ }$

Q.

Which of the following is/are true about kite?

i) Kite is a type of parallelogram.

ii) The diagonals of a kite are perpendicular to each other.

iii) The diagonals of a kite bisect each other.

iv) One pair of opposite angles is equal to each other.

1. i, ii, iii

2. ii, iv

3. ii, iii, iv

4. All of these

Q.

The perimeter of two similar triangles $∆ABC$ and $∆PQR$ are $60cm$ and $36cm$ respectively. If $PQ=9cm,$then find the value of $AB.$

Q.

The area of the $∆ABC$, in which $a=1,b=2,\angle C={60}^{\circ }$, is

1. $4squnits$

2. $\frac{1}{2}squnits$

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

4. $\sqrt{3}squnits$

Q. If two sides are equal the third side would also be equal

Q.

If two sides of a triangle are given, then the area of the triangle will be maximum if the angle between the given sides be.

1. $\frac{\mathrm{\pi }}{3}$

2. $\frac{\mathrm{\pi }}{4}$

3. $\frac{\mathrm{\pi }}{6}$

4. $\frac{\mathrm{\pi }}{2}$

Q.

Assertion : In $\Delta ABC$, median AD is produced to X such that AD = DX. Then, ABXC is a parallelogram.

Reason : Diagonals AX and BC bisect each other at right angles.

Which of the following is correct?

1. If both assertion and reason are true and reason is the correct explanation of assertion

2. If both assertion and reason are true but reason is not the correct explanation of assertion

3. If assertion is true but reason is false

4. If assertion is false but reason is true

Q.

If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.

Q.

In a quadrilateral $ABCD$, $\angle B=90°$, $A{D}^2=A{B}^2+B{C}^2+C{D}^2$, prove that $\angle ACD=90°$.

Q.

In the given figure, find $PM$.

1. $2cm$

2. $3cm$

3. $4cm$

4. $5cm$

Q.

If $∆=a^2-{(b-c)}^2$, where $∆$ is the area of $∆ABC$, then $\tan A$ is equal to

1. $\frac{15}{16}$

2. $\frac{8}{17}$

3. $\frac{8}{15}$

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

Q. Find the value of x.

1. ${60}^{\circ }$

2. ${30}^{\circ }$
3. ${120}^{\circ }$
4. ${90}^{\circ }$

Q.

Find the value of $\sin {15}^{\circ }$using $\sin {30}^{\circ }$.

Q.

If$\cos 20°-\sin 20°=p$then $\cos 40°$is equal to

1. $p^2\sqrt{(2–p^2)}$

2. $p\sqrt{(2–p^2)}$

3. $p+\sqrt{(2–p^2)}$

4. $p–\sqrt{(2–p^2)}$

Q.

If the non zero numbers $x,y,z$ are in AP and ${\tan }^{-1}x,{\tan }^{-1}y,{\tan }^{-1}z$ are in AP then

1. $x=y=z$

2. $xy=yz$

3. $x^2=yz$

4. $z^2=xy$

Q.

How do you do inverse cosine?

Q.

Let $x,𝑦$and $z$ be positive real numbers. Suppose $𝑥,𝑦$ and $z$ are the lengths of the sides of a triangle opposite to its angles $𝑋,𝑌$and $Z$, respectively. If

$\begin{array}{l}tan\frac{X}{2}+tan\frac{Z}{2}=\frac{2y}{x+y+z}\end{array}$

then which of the following statements is/are TRUE

1. $2𝑌=𝑋+𝑍$

2. $Y=𝑋+𝑍$

3. $\begin{array}{l}\tan \frac{\mathrm{X}}{2}=\frac{\mathrm{x}}{\mathrm{y}+\mathrm{z}}\end{array}$

4. ${x^2}^{}+z^2-y^2=xz$