Ratio of Sides of a Triangle Determined Using Its Angles

Q.

In a given $\Delta$ ABC, AB =20 cm, $\angle A={30}^{\circ }$ as shown in figure. If the area of the triangle is $100c{m}^2$, then the length of AC is equal to

1. $10\sqrt{3}cm$

2. $20\sqrt{3}cm$

3. 10 cm

4. 20 cm

Q.

An observer from the top of lighthouse observes a boat speeding away from the tower. When the boat is at a distance of 100 m from the feet of the tower, the boat makes an angle depression $45°$ with the mans eye. After 10 seconds, the angle of depression becomes $30°$. Assuming the water to be still, calculate the speed of the boat.

1. $(\sqrt{3}-1)m/sec$

2. $10(\sqrt{3}-1)m/sec$

3. $\frac{(\sqrt{3}-1)}{10}m/sec$

4. $20(\sqrt{3}-1)m/sec$

Q.

In the given right-angled triangle the angle with $\angle B={30}^{\circ }$, the ratio of the sides AB : AC: BC will be equal to

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

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

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

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

Q.

A rhombus whose side measures 40 cm has one of its angles as ${60}^{\circ },$, then the length of its diagonals are

1. 40 cm, 80 cm

2. $40\sqrt{3}cm,80cm$

3. $40cm,40\sqrt{3}cm$

4. $80cm,80\sqrt{3}cm$

Q.

A rhombus having each side 10 cm has one of its angles as ${120}^{\circ }$, then the area of the rhombus will be equal to

1. $100\sqrt{3}c{m}^2$

2. $50\sqrt{3}c{m}^2$

3. $150\sqrt{3}c{m}^2$

4. $75\sqrt{3}c{m}^2$

Q.

In a right $△$ ABC, if $\angle$ A is acute and tanA = 3/4, find the remaining trigonometric ratios of $\angle$ A.

Q. In a right angled triangle ABC right angled at B, $5\times sinA=3$.
Find the value of $cosC+tanA+cosecC.$

1. $\frac{12}{5}$
2. $\frac{13}{5}$
3. $\frac{4}{5}$
4. $\frac{13}{12}$

Q.

In triangle $ABC,\angle B={30}^{\circ },\angle C={45}^{\circ }$. If CD = 8 cm, then AB is equal to

1. 8 cm

2. 16 cm

3. $8\sqrt{3}cm$

4. $16\sqrt{3}cm$

Q.

In triangle ABC, $\angle B={30}^{\circ }$ and $\angle C={45}^{\circ }$. If AB =4 cm, then find the length of CD.

1. 2 cm

2. 4 cm

3. $2\sqrt{3}cm$

4. $4\sqrt{3}cm$

Q.

In the rhombus ABCD, Side AD = 12 cm and $\angle B$ is ${135}^{\circ }$ as shown in figure. The height (DE) and the area of the rhombus will be equal to

1. $12\sqrt{2}cm,36c{m}^2$

2. $6\sqrt{2}cm,36\sqrt{2}c{m}^2$

3. $6\sqrt{2}cm,72\sqrt{2}c{m}^2$

4. $12\sqrt{2}cm,72c{m}^2$

Q. Using Theorem $6.1,$ prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Q.

The length of the diagonal AC of a rhombus ABCD is 20 cm and one of its angle measures ${120}^{\circ },$ then the length of the other diagonal BD is

1. 10 cm

2. 20 cm

3. $10\sqrt{3}cm$

4. $\frac{20}{\sqrt{3}}cm$

Q.

To have a steep slide at a height of 6 m, and inclined at an angle of ${60}^{\circ }$ to the ground. What should be the length of the slide (in m)?

1. 3

2. 5

3. 4

4. 6

Q.

In $\Delta ABC$, which is circumscribed by a circle , $\angle A={45}^{\circ }$, BC =6 cm. The diameter of the circumcircle is equal to

1. $3\sqrt{2}cm$

2. $6\sqrt{2}cm$

3. 6 cm

4. 3 cm

Q.

AC and BC are two equal chords of a circle with diameter AB forming a $\Delta ABC$ as shown in the figure. If the equal chords have length 20 cm, then the area of the circle is equal to

1. $50\pi c{m}^2$

2. $100\pi c{m}^2$

3. $200\pi c{m}^2$

4. $150\pi c{m}^2$

Q.

In Triangle $ABC,AB=15cm,AC=8cm,\angle A={50}^{\circ },$ as shown below.The length of BC will be equal to $[sin{50}^{\circ }=0.76,cos{50}^{\circ }=0.64]$

1. 10.2 cm

2. 13.9 cm

3. 12.5 cm

4. 11.6 cm

Q. ABCDis a rectangle in which AE line makes 30 degree with angle A which meet DC at E.find AE and EC.

Q.

In Triangle $ABC,AB=15cm,AC=8cm,\angle A={50}^{\circ },$ as shown below. Find the length of BC. $[sin{50}^{\circ }=0.76,cos{50}^{\circ }=0.64]$

Q.

AC and BC are two equal chords of a circle with diameter AB forming a $\Delta ABC$ as shown in the figure. If the radius of the circle is 5 cm. find the length of the equal chords.

1. $10\sqrt{2}cm$

2. $5\sqrt{2}cm$

3. 5 cm

4. 10 cm

Q.

The perimeter and area of the rectangle whose 12 cm long diagonal makes an angle of ${30}^{\circ }$ with one of its longer sides will be equal to

1. $18+12\sqrt{3},18\sqrt{3}c{m}^2$

2. $12+12\sqrt{3},36\sqrt{3}c{m}^2$

3. $12+18\sqrt{3},24\sqrt{3}c{m}^2$

4. $9+18\sqrt{3},27\sqrt{3}c{m}^2$

Q.

One angle of a $\Delta ABC$ which is circumscribed by a circle, has one of the angles as ${135}^{\circ }$ and its opposite side is 5 cm as shown in the figure. The diameter of its circumcircle is equal to

1. $10\sqrt{2}cm$

2. $20\sqrt{2}cm$

3. $5\sqrt{2}cm$

4. $5\sqrt{3}cm$

Q.

In the given right angle triangle ABC, if $\alpha ={30}^{\circ }$ and AC = 10 cm then find the value of the side BC in cm.

1. 5

2. 3

3. 1

4. 2

Q.

If $\alpha \mathrm{and}\beta$ are the complementary angles, then what is $\sqrt{\mathrm{cosec}\alpha ·\mathrm{cosec}\beta }{\left(\frac{\sin \alpha }{\sin \beta }+\frac{\cos \alpha }{\cos \beta }\right)}^{-1/2}$ equal to:

1. $0$

2. $1$

3. $2$

4. None of these

Q.

In the given triangle ABC, $\angle A={45}^{\circ }$, BC = 9 cm as shown in the figure. The diameter of the circumcircle is equal to

1. $6\sqrt{2}cm$

2. $9\sqrt{2}cm$

3. 18 cm

4. 9 cm

Q.

A rhombus of side 18 cm has one of its angles as ${135}^{\circ }$, then the area of the rhombus will be equal to

1. $81c{m}^2$

2. $162\sqrt{2}c{m}^2$

3. $81\sqrt{2}c{m}^2$

4. $162c{m}^2$

Q.

One angle of a $\Delta ABC$ is ${150}^{\circ }$ and its opposite side is 3 cm as shown in the figure. The diameter of its circumcircle is equal to

1. 6 cm

2. $6\sqrt{2}cm$

3. $6\sqrt{3}cm$

4. $3\sqrt{3}cm$

Q.

In a triangle ABC, $\angle A={40}^{\circ },\angle B={80}^{\circ },AB=8cm.$ as shown in figure. The length of the two sides AC $\&$ BC is equal to $\left[\begin{array}{c}sin{40}^{\circ }=0.64\\ sin{60}^{\circ }=0.86\\ sin{80}^{\circ }=0.98\end{array}\right]$

1. 6.45 cm, 8.5 cm

2. 5.5 cm, 8.2 cm

3. 5.95 cm, 9.11 cm

4. 6.5 cm, 9.5 cm

Q.

In triangle $ABC,\angle A=\angle B={30}^{\circ },AB=18cm$ as shown in figure. The perimeter of the triangle will be to equal to

1. $(18\sqrt{3}+12)cm$

2. $(12\sqrt{3}+18)cm$

3. $12\sqrt{6}+18cm$

4. $18\sqrt{6}+12cm$

Q.

In the given figure, a circle is circumscribing $\Delta ABC$ where $\angle A={125}^{\circ }$ and side BC=8cm. The diameter of the circumcircle is equal to

$[sin{55}^{\circ }=0.82]$

1. 12.5 cm

2. 10.75 cm

3. 9.75 cm

4. 8.25 cm

Q.

In right-angled triangle ABC, $\angle A={45}^{\circ }$, The ratio of AB : AC : BC will be equal to

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

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

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

4. $1:1:1$