Finding Ratio of Sides in Triangle When Angles Are Given

Q.

The vector $\hat{i}+x\hat{j}+3\hat{k}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4\hat{i}+(4x-2)\hat{j}+2\hat{k}$. The values of $x$ are?

1. $\left\{-\frac{2}{3},2\right\}$

2. $\left\{\frac{1}{3},2\right\}$

3. $\left\{\frac{2}{3},0\right\}$

4. $\left\{2,7\right\}$

Q.

The distance between the foci of the conic $7x^2-9y^2=63$is equal to

1. $8$

2. $4$

3. $3$

4. $7$

5. $12$

Q. In $△XYZ,$ XY = 4 cm, YZ = 6 cm, XZ = 5 cm, If $△XYZ\sim △PQR$ and PQ = 8 cm then find the lengths of remaining sides of $△PQR.$

Q. A ladder is placed in such a way that it's foot is at a distance of 12m from the wall and it's top reaches a window 5m above the ground. Find the length of the ladder.

Q.

In a right angled triangle, the sides are $a,b\mathrm{and}c$ with $c$ as hypotenuse and $c-b\ne 1,c+b\ne 1$. Then, the value of $\frac{({\log }_{c+b}a+{\log }_{c-b}a)}{\left(2{\log }_{c+b}a\times {\log }_{c-b}a\right)}$ will be:

1. $2$

2. $-1$

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

4. $1$

Q.

In a triangle ABC, angle A=90°, CA=AB and D is a point on AB produced. Prove that

DC² - BD² = 2AB × AD

Q.

If two sides of a triangle are 8 cm and 13 cm, then the length of the third side is between a cm and b cm. Find the values of a and b such that a is less than b.

Q.

The sides forming the right angles and hypotenuse of a right angled triangle are xcm, (3x-2)cm and (2x+2)cm respectively. Calculate the length of the sides of the triangle.

Q. In a right angled triangle , one acute angle is double the other. Prove that hypotenuse is double the smallest side.

Q.

What is the ratio between AE and AF if EF = 1m ?

1. 1 : $\sqrt{5}$

2. 2 : $\sqrt{5}$

3. 1 : $\sqrt{3}$

4. $\sqrt{5}$ : 3

Q.

What is the length of BC in the given figure?

1. $2\sqrt{2}$ units

2. 4 units

3. $\sqrt{2}$ units

4. $4\sqrt{2}$ units

Q.

If the lengths of two sides of a triangle are 3 and 4 respectively, then which one of the following is true about the third side k?

1. 3+4 >k

2. 3+4 =k

3. 3+4 <k or 3+4 =k

4. 3+4 <k

Q.

What is the Ratio between KQ and KM? PQRS is a square and K and M are midpoints of PQ and QR respectively.

1. 1 : 2

2. 1 : $\sqrt{2}$

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

4. 2 : $\sqrt{2}$

Q.

In a right-angled triangle, if the lengths of the two legs are 1 m each, the length of the hypotenuse is a whole number.

1. True

2. False

Q. If 15 cot N = 8, find the value of $\frac{1}{\sqrt{se{c}^{2}N-1}}$

Q.

What is the length of AD in the given figure ABCD? ( Take $\sqrt{2}$ = 1.4 and $\sqrt{5}$ = 2.2

1. $\sqrt{3}$ m

2. $\sqrt{2}$ m

3. $\sqrt{5}$ m

4. $\sqrt{7}$ m

Q.

PCB is a right angled triangle. Find AB : PC, given BC = 4 cm.

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

2. $\sqrt{2}$ : 2

3. $\sqrt{2}$ : 1

4. 1 : $\sqrt{2}$

Q.

The distance cosines of a line are $\left(\frac{1}{c},\frac{1}{c},\frac{1}{c}\right)$, then

1. $0<c<1$

2. $c>2$

3. $c=\pm \sqrt{2}$

4. $c=\pm \sqrt{3}$

5. $c=\pm 3$

Q.

If$f\left(x\right)={\int }_0^e{e}^{\frac{-x^2}{2}}(1-t^2)dt$, minimum at $x=?$

1. $1$

2. $-1$

3. $2$

4. $-2$

Q.

In the given figure, if $\angle$ACB = ${30}^{\circ }$, what is the value of $\angle$ACD in degrees?

___

Q.

What Will be the height (AO) of the triangle if $\angle$ ABO given is changed from ${45}^{\circ }$ to ${60}^{\circ }$?

1. $\sqrt{3}$

2. $2\sqrt{3}$

3. $3\sqrt{3}$

4. $\sqrt{6}$

Q. A right triangle has two sides of lengths 4 m and 4 m respectively. The length of its hypotenuse is.

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

Q.

What is the ratio between length of perpendicular AO and the perimeter of the $△$ABC?

1. 1 : 2

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

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

4. 1 : (2 + $\sqrt{2}$)

Q. Let us define a statement as a sentence that can be judged to be true or false.
Which of the following is not a statement?
(a) 3 + 5 = 7.
(b) Kunal is a tall boy.
(c) The sum of the angles of a triangle is 90°.
(d) The angles opposite to equal sides of a triangle are equal.

Q.

Anand and Amir ran 35 km 750 m and 28 km 90 m, respectively in two hours. Who ran more and by how many kilometers?

Q. Mark (✓) against the correct answer in each of the following:
In an isosceles right triangle, the length of the hypotenuse is $4\sqrt{2}\mathrm{cm}$.
(a) $4\sqrt{3}\mathrm{cm}$
(b) 6 cm
(c) 5 cm
(d) 4 cm
Figure

Q.

A right triangle has two sides of lengths 4 m and 4 m respectively. Find the length of the hypotenuse by using ratio of sides concept.

1. 2$\sqrt{2}$m

2. 4$\sqrt{3}$m

3. 4$\sqrt{2}$m

4. 3$\sqrt{2}$m

Q.

If we join the points A and P, What will be the length of AP if BC = 2 cm.

1. $2\sqrt{2+\sqrt{2}}$ cm

2. $4\sqrt{4+\sqrt{2}}$ cm

3. $2\sqrt{4+\sqrt{8}}$ cm

4. $2\sqrt{1+\sqrt{2}}$ cm

Q.

Find the length of AC in the given figure, if the sides are in the ratio $1:\sqrt{3}:2$

1. $\sqrt{2}$ metres

2. $\sqrt{5}$ metres

3. 1 metre

4. $\sqrt{3}$ metres

Q. The perpendicular sides of a right angle isosceles triangle is 2acm .Find its hypotenuse?