A flagpole is supported by three wires B A , B C and B D , as shown in this diagram. Show that the triangle A B C is not right angled.

Consider ∆ A D B Applying cosine rule, cos ( A D ^ B ) = A D 2 + B D 2 - A B 2 2 A D · B D ⇒ cos ( A D ^ B ) = 12 2 + 20 2 - 28 2 2 ( 12 ) ( 20 ) = 144 + 400 - 784 480 ⇒ cos ( A D ^ B ) = - 240 480 = - 1 2 ⇒ A D ^ B = 120 ° ⇒ B D ^ C = 180 ° - A D ^ B = 60 ° Consider, ∆ B C D From sine rule, sin C B D = sin D B C ⇒ sin C 12 = sin 60 ° 13 ⇒ sin C = 12 13 3 2 ⇒ sin C = 0 . 799 ≅ 0 . 8 ⇒ ∠ C = 53 · 1 ° Consider ∆ A B C Applying sine rule, sin A B C = sin C A B ⇒ sin A 13 = 0 . 8 28 ⇒ sin A = 0 . 371 ⇒ ∠ A = 21 . 7 ° From angle sum property ∠ A + ∠ B + ∠ C = 180 ° ⇒ ∠ B = 180 ° - 53 · 1 ° + 21 . 7 ° ⇒ ∠ B = 105 . 2 ° ∴ A B C is not a right angled triangle. Hence, shown.

Without actually calculating the size of the angle you will be able to say that A B ^ C is obtuse using cos ( A B ^ C ) = - 11 24 , explain why you are able to do this.

Given, cos A B ^ C = - 11 24 Without calculating A B ^ C , we can say that A B ^ C is obtuse, because we know that cosine is negative in second and fourth quadrants. ⇒ A B ^ C ∈ 90 ° , 180 ° or 270 ° , 360 ° But an angle of triangle cannot be ≥ 180 ° Therefore A B ^ C ∈ 90 ° , 180 ° If an angle is > 90 ° , then the angle is called an obtuse angle. Hence, we can say that A B ^ C is an obtuse without calculating.

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A flagpole is supported by three wires $B A$ , $B C$ and $B D$ , as shown in this diagram.

Find the angle of elevation of the top of the tower from point $C$ .

Important Questions on Relationships in Space: Geometry and Trigonometry in 2D and 3D

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IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 4

A flagpole is supported by three wires $B A$ , $B C$ and $B D$ , as shown in this diagram.

Question Image

Show that the triangle $A B C$ is not right angled.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 5

A dockyard crane $A B C$ , is used to unload cargo from ships. $A B = 22.5 m, A C = 48 m$ and angle $A B C = 46 °$ . Find $A \hat{C} B$ .

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MEDIUM

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 5

A dockyard crane $A B C$ , is used to unload cargo from ships. $A B = 22.5 m, A C = 48 m$ and angle $A B C = 46 °$ . Find the length of the lifting arm, $B C$ .

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IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 7

Two security cameras are positioned on the ceiling of a gym, $10 m$ apart. One camera has an angle of depression of $50 °$ to a point on the floor, and the other camera has an angle of depression of $60 °$ to the same point. Calculate the height, $h$ metres, of the gym.

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HARD

Diploma

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 10

A ship leaves Shanghai on a bearing of

030 °

. It sails a distance of

500 km

to Jeju. At

B

, the ship changes direction to a bearing of

100 °

. It sails a distance of

320 km

to reach Nagasaki. Calculate the bearing and distance for the ship to return from Nagasaki to Shanghai.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 12

Calculate the value of $\cos (A \hat{B} C)$ in this triangle.

Question Image

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Exercise 11I>Q 12

Without actually calculating the size of the angle you will be able to say that

A \hat{B} C

is obtuse using

\cos (A \hat{B} C) = \frac{- 11}{24}

, explain why you are able to do this.

MEDIUM

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 11 - Relationships in Space: Geometry and Trigonometry in 2D and 3D>Chapter review>Q 1

Find the volume and the surface area of a square-based pyramid with a base side length of $8 m$ and a height of $3 m$ .

Question Image

A flagpole is supported by three wires BA, BC and BD, as shown in this diagram. Find the angle of elevation of the top of the tower from point C.

Important Questions on Relationships in Space: Geometry and Trigonometry in 2D and 3D

Learn and Prepare for any exam you want !

A flagpole is supported by three wires $B A$ , $B C$ and $B D$ , as shown in this diagram.

Find the angle of elevation of the top of the tower from point $C$ .