LABELLING RIGHT ANGLED TRIANGLES

Problem 1:

In the diagrams below, name the:

i) Hypotenuse

ii)  Side opposite the angle marked θ

iii) Side adjacent to the angle marked θ.

a.

Solution :

i) Hypotenuse = AB (opposite to 90 degree)

ii)  Side opposite the angle marked θ = BC

iii) Side adjacent to the angle marked θ = AC

Problem 2 :

Solution :

i) Hypotenuse = RQ

ii) Side opposite the angle marked θ = PR

iii)  Side adjacent to the angle marked θ = PQ

Problem 3 :

Solution :

i) Hypotenuse = XZ

ii) Side opposite the angle marked θ = YZ

iii)  Side adjacent to the angle marked θ = XY

Problem 4 :

The right angled triangle alongside has hypotenuse of length a units and other sides of length b units and c units. θ and Φ are the two acute angles. Find the length of the side:

a)   Opposite θ

b)   Opposite Φ

c)   Adjacent to θ

d)   Adjacent to Φ

Solution :

a)   Opposite θ = b

b)   Opposite Φ = c

c)   Adjacent to θ = c

d)   Adjacent to Φ = b

Problem 5 :

Given the right angle triangle shown here:

state the length of :

1) the side opposite angle a

2) the side opposite angle b

3) the hypotenuse

4) the side adjacent to angle b 

5) the side adjacent to angle a

Solution :

1) the side opposite angle a = 4

2) the side opposite angle b = 3

3) the hypotenuse = 5

4) the side adjacent to angle b = 4  

5) the side adjacent to angle a = 3

Problem 6 :

Given the right angle triangle ABC shown here

name the:

1) Hypotenuse

2) Adjacent side to b

3) Opposite side to a

4) Opposite side to b

5) Adjacent side to a

Solution :

1) Hypotenuse = AB

2) Adjacent side to b = BC

3) Opposite side to a = BC

4) Opposite side to b = AC

5) Adjacent side to a = AC

Problem 7 :

You are measuring the height of a spruce tree. You stand 45 feet from the base of the tree. You measure the angle of elevation from the ground to the top of the tree to be 59°. Find the height h of the tree to the nearest foot.

Solution :

Opposite side = h ft, adjacent side = 45 ft

tan θ = Opposite side / Adjacent side

tan 59 = h/45

1.6642 = h/45

h = 1.6642(45)

h = 74.88 ft

Approximately 75 ft.

Problem 8 :

The backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. Use the Pythagorean Theorem to approximate the distance between the rods where they meet the backboard.

Solution :

Hypotenuse = 13.4 inches

Using Pythagorean theorem,

13.4² = x² + 9.8²

179.56 = x² + 96.04

x² = 179.56 - 96.04

x² = 83.52

x = √83.52

x = 9.13 ft

Problem 9 :

The fire escape forms a right triangle, as shown. Use the Pythagorean Theorem to approximate the distance between the two platforms.

Solution :

Hypotenuse = 16.7 ft

Using Pythagorean theorem,

16.7² = x² + 8.9²

278.89 = x² + 79.21

x² = 278.89 - 79.21

x² = 199.68

x = √199.68

x = 14.13 ft

Problem 10 :

The top of the slide is 12 feet from the ground and has an angle of depression of 53°. What is the length of the slide?

Solution :

θ = 53

Opposite side = 12 ft

Length of the slide = Adjacent side

tan θ = Opposite side / Adjacent side

tan 53 = 12/Length of slide

1.327 = 12/Length of slide

Length of slide = 12/1.327

= 9.04 ft

Problem 11 :

The angle between the bottom of a fence and the top of a tree is 75°. The tree is 4 feet from the fence. How tall is the tree? Round your answer to the nearest foot.

Solution :

θ = 75

Opposite side = x

Adjacent side = 4 ft

tan θ = Opposite side / Adjacent side

tan 75 = x/4

3.732 = x/4

x = 4(3.732)

x = 14.928

Approximately 15 ft.

Problem 12 :

An anemometer is a device used to measure wind speed. The anemometer shown is attached to the top of a pole. Support wires are attached to the pole 5 feet above the ground. Each support wire is 6 feet long. How far from the base of the pole is each wire attached to the ground?

Solution :

Hypotenuse² = Opposite side² + Adjacent side²

6² = d² + 5²

36 = d² + 25

d² = 36 - 25

d² = 11

d = √11

= 3.31 ft

So, the required distance is 3.31 ft.