Apollonius's Theorem

Q.

In$∆PQR,\angle R=\frac{\mathrm{\pi }}{2}$, if $\tan \left(\frac{P}{2}\right)$ and $\tan \left(\frac{Q}{2}\right)$ are the roots of $a{x}^2+bx+c=0,a\ne 0$, then

1. $b=a+c$

2. $b=c$

3. $c=a+b$

4. $a=b+c$

Q.

BL and CM are medians of a triangle ABC right angled at A. Prove that
$4({BL}^2+{CM}^2)=5{BC}^2$. [3 MARKS]

Q.

If AD, BE and CF are the medians of an equilateral triangle ABC, then which of the following statements is valid?

1. AB² + BC² + AC² = AD² + BE² + CF²

2. 2 (AB² + BC² + AC²) = 3 (AD² + BE² + CF²)

3. 3 (AB² + BC² + AC²) = 4 (AD² + BE² + CF²)

4. AB² + BC² + AC² = 3 (AD² + BE² + CF²)

Q.

Four rods each of length $I$ have been hinged to form a rhombus. Vertex $A$ is fixed to a rigid support, vertex $C$ is being moved along the $x-$axis with a constant velocity $v$ as shown in the figure. The rate at which vertex $B$ is approaching the $x-$axis at the moment the rhombus is in the form of a square is

Q.

Let $ABC$be a triangle such that $\angle ACB=\frac{\pi }{6}$ and let $a,b$ and $c$ denote the lengths of the sides opposite to$A,B$ and $C$ respectively. The value(s) of $x$ for which $a=x^2+x+1,b=x^2-1$and $c=2x+1$is (are)

1. $-(2+\sqrt{3})$

2. $1+\sqrt{3}$

3. $2+\sqrt{3}$

4. $4\sqrt{3}$

Q.

Two poles of height $6\mathrm{m}$ and $11\mathrm{m}$ stands on a plane ground, if the distance between their feet is $12\mathrm{m}$. Find the distance between their tops.

Q.

In fig., DE || BC. If AD = x, DB = x - 2, AE = x + 2 and EC = x - 1, find the value of x.

Q.

If the $1^{st}$ term and common ratio of a G.P. are $1$ and $2$ respectively then $S_1+S_3+S_5+...+S_{2n-1}=$

1. $\left(\frac{1}{3}\right)\left(2^n-5n+4\right)$

2. $\left(\frac{1}{3}\right)\left(2^{2n+1}-5n\right)$

3. $\left(\frac{1}{3}\right)\left(2^{2\mathrm{n}+1}-5{\mathrm{n}}^2\right)$

4. $\left(\frac{1}{3}\right)\left(2^{2n+1}-3n-2\right)$

Q.

In the figure given below AB = 3 cm, AC = 5 cm and AD = 4 cm and D is the midpoint of BC. Then the length of BD is :

1. 4 cm

2. 1 cm

3. 2 cm

4. 3 cm

Q.

State and prove Pythagoras theorem

Q.

$△ABC$ is an isosceles right triangle, right angled at $\mathrm{C}.$ Prove that $\mathrm{AB}{}^2=2{\mathrm{AC}}^2$

Q.

In $△$ABC, the medians BE and CF intersect at G. AD is a line meeting BC at D. If GD is 1.5 cm, then find the value of AD.

1. 2.5 cm
2. 3 cm
3. 4 cm
4. 4.5 cm

Q.

If $u$ and $w$ are two mutually perpendicular unit vectors and $u=av+bw,$where $a$ and $b$ are non-zero real numbers, then the angle between $u$and $w$ is?

1. ${\cos }^{-1}\left(a\right)$

2. ${\cos }^{-1}\left(b\right)$

3. ${\cos }^{-1}\left(\frac{a}{\sqrt{a^2+b2}}\right)$

4. ${\cos }^{-1}\left(\frac{b}{\sqrt{a^2+b2}}\right)$

Q. In ΔABC, AB = 11 cm, AC = 17 cm, BC = 12 cm. Find the length of the median drawn from point A to BC.

1. 13 cm
2. 15 cm
3. 9 cm
4. 10 cm

Q. In $△ABC$, if $AD$ is the median, show that $A{B}^2+A{C}^2=2(A{D}^2+B{D}^2)$.

Q.

Which of the following theorems is a prerequisite for proving Apollonius Theorem?

1. Euler's Formula

2. Guass-Markov Theorem

3. Wilson's Theorem

4. Pythogoras Theorem

Q. In $△ABC$, seg AQ is the median on BC. $A{B}^2+A{C}^2=122$, BC=$10$ units. Find the length of the median on side BC.

Q. In an equilateral $\Delta ABC,$ D is a point on side BC such that BD $=\frac{1}{3}$BC. Prove that $9(AD{)}^2=7(AB{)}^2$.
OR
Prove that, in a right triangle, the square on the hyporenuse is equal to the sum of the squares on the other two sides.

Q.

Use the diagram below to approximate the length of diagonal to the nearest tenth. Explain your method.

Q. In a $\Delta ABC$, $AB=4$ cm and $AC=8$ cm. If M is the mid point of BC and $AM=3$ cm, then length of BC, in cm, is:

1. $2\sqrt{2}6$
2. $2\sqrt{3}1$
3. $\sqrt{3}1$
4. $\sqrt{2}6$

Q. In the given figure, $△ABC={90}^o$ and $BM$ is a median, $AB=8cm$ and $BC=6cm$. Then, length $BM$ is equal to:

1. $4cm$
2. $3cm$
3. $5cm$
4. $7cm$

Q. In a $\Delta$ $ABC,AC=4,BC=2,AB=6$, find the value of $AD$. (Use Apollonius theorem).

1. 2
2. 3
3. 4
4. 5

Q. In a $△ABC$, $AB=4$ cm and $AC=8$ cm. If M is the midpoint of BC and $AM=3$ cm, then the length of $BC$ in cm is:

1. $2\sqrt{26}$
2. $2\sqrt{31}$
3. $\sqrt{31}$
4. $\sqrt{26}$

Q. In $△PQR$, M is the midpoint of side QR. If PQ is $11$ cm, PR is $17$ cm and QR = $12$ cm, find PM.

1. $15$ cm
2. $13$ cm
3. $17$ cm
4. $23$ cm

Q. In $ABC$, the lengths of sides $BC,CA$ and $AB$ are $a,b$ and $c$ respectively. Median $AD$ drawn from $A$ is perpendicular to $AB$, then $b$ is:

Q.

The perimeter of the rectangle whose length is $60cm$and a diagonal is $61cm$ is

1. $120cm$

2. $122cm$

3. $71cm$

4. $142cm$

Q.

In the given triangle , which of the following is true ?

1. PQ² + QR² = PR²

2. All of these

3. PQ² + PR² = QR²

4. PR² + QR² = PQ²

Q. In $△ABC,{AB}^2+{BC}^2=122,BC=10$. Find the length of the median on side $BC$.

Q.

In the given figure, the length of $AB$, if $\mathrm{OA}=5\mathrm{cm}$ and $\mathrm{OC}=3\mathrm{cm}$ is

1. $4cm$

2. $5cm$

3. $7cm$

4. $8cm$

Q.

In the figure, $AC=10cm,PC=15cm,AB=8cm,$ find PB.

1. $8cm$

2. $6cm$

3. $9cm$

4. $7cm$