### ■

**apply Pythagoras' theorem to solve problems (ACMEM116)**

### ■

**apply the tangent ratio to find unknown angles and sides in right-angled triangles (ACMEMl 17)**

### ■

**work with the concepts of angle of elevation aiid angle of depression (ACMEMl 18)**

**Ill ** **apply the cosine and sine ratios to find unknown angles and sides in right-angled triangles (ACMEM119)**

### ■

**solve problems involving bearings. (ACMEM120) �**

### How are we ever going to use this?

• When we have to determine a length we can't measure

• When we need to check that the angles in a quadrilateral are 90 degrees

• Many people who work in trades, for example, carpenters and builders, use right-angled triangle calculations in their work

**t1111 ****PYTHAGORAS' THEOREM**

Pythagoras lived in Greece from 580 to 500 BCE. He was the

founder of a secret brotherhood whose members were interested in the mystical properties of numbers. The famous theorem that relates the sides of a right-angled triangle carries his name even though no one is sure whether it was Pythagoras himself, or another member of the brotherhood, who first proved the

theorem. What we do know is that humans knew about the Pythagorean relationship well before Pythagoras' time. Pythagoras' theorem states that in a right-angled triangle:

*C *

Side 2

The square of the hypotenuse is equal to the sum of the squares of the other two sides.
(hypotenuse/= (side 1)**2 **_{+ (side 2)}**2 **

Sometimes we write the formula as *a2 *_{= }
*b2 *+ *c2 *

**156 I NELSON SENIOR MATHS **Essentials 12

* b *
Side 1

### 0 Example 1

Use Pythagoras' theorem to find the height *(h) *_{of this roof truss. Answer correct to 1 decimal }
place.

**/4****\ **

### \�

### 1---

5.8 m### ---1

### Solution

The hypotenuse is 6.35 and one side is 5.8.

6.35**2 **_{= }_{5.8}**2 **_{+ }* _{h}2 *
6.35

**2**

**-**5.8

**2**=

*h*

**2***h2*

_{=6.6825 }

*h=*

_{✓6.68,25 }= 2.585 "'2.6 m to 1 decimal place

We can also use Pythagoras' theorem to test whether a triangle contains a right-angle. If the

theorem works, there's a right-angle. If the theorem doesn't work, the angle isn't 90°_{. }

### 0 Example 2

Is this triangle right-angled?

### Solution

We need to check whether 56**2 **_{= 36}**2 **_{+ 44}**2 _{. }**

If both sides of the equation are equal, then
there is a right angle. If they aren't equal, there
is no right angle.
56
36
44
56**2 **_{= }_{36}**2 **_{+ 44}**2 _{• }**
3136 = 1296 + 1936
3136 = 3232

This statement is false.

Pythagoras' theorem doesn't work. The
triangle is *not *_{right}_{-}_{angled. }

**EXERCISE 7 .01 **

_{Pythagoras' theorem }

_{Pythagoras' theorem }

**•@j**

**,,**

**jciJI **

The diagram represents a wheelchair ramp. The 36 m long ramp covers a horizontal
distance of 35 m. Use Pythagoras' theorem to calculate the rise *h *m, of the ramp. Answer

correct to 1 decimal place.
*�hm *

35m

2 What length of wire is required to connect the top of a 23-metre TV antenna to a hook that is 6 metres from the base of the antenna? Answer correct to 1 decimal place.

3 Stuart installs a new section of pipe to join two existing pipes.

Calculate the length of the new section of pipe. Express your answer in metres, correct to the nearest millimetre.

6m I I I I 23m :sm I I I

When a measurement is in metres, the nearest millimetre means three decimal places. The nearest centimetre is two decimal places.

### a:=:===�� ----D

3m4 Ivan is flying a small plane at 160 km/h against a

50 km/h wind, as shown in the diagram. Calculate the plane's ground speed, correct to the nearest km/h.

**158 I NELSON SENIOR MATHS Essentials 12**

Wind speed SO km/h

Speed of the plane in the air 160 km/h

Plane's ground speed

5

**iil•ini•i?J **

Use Pythagoras' theorem to test whether each triangle is right-angled.
a
### b {�

7.5### �

**d**� 4.5 28 24

6 John is laying a concrete slab 12 m by 3 m in front of his shed. He uses Pythagoras' theorem to check that the corners of the slab are right angles. How long should the diagonal be? Express your answer in metres, correct to the nearest centimetre.

20

1200m

7 The school cross-country course is in the shape of a right-angled triangle. The first leg is 1200 m and the second leg is 950 m. The third leg through thick scrub is difficult to measure. Calculate its length, correct to

the nearest metre.

### ,,

Thirdleg ,

8 Belle had to cross the Paroo River to get from Mud Springs to Eulo. The new bridge across the river was closed for repairs, so she had to use the old bridge. How much further did she have to travel using the old road and old bridge compared to the direct route across the new bridge? Answer correct to one decimal place.

Eulo

950m

**Pythagoras' **
**theorem time **
**trial **

9 David is the pilot of a small plane. He planned to fly 200 km from Mount Surprise to Cairns. Because of poor weather conditions between Mount Surprise and Cairns, David flew 135 km due north to Chillagoe, then turned due east and flew to Cairns. Calculate the straight line distance from Chillagoe to Cairns, correct to the nearest kilometre.

135km

Mount Surprise

N

### +

**10 ** The h_{yp}otenuse of a right-angled triangle is 24 m. Draw two possible triangles, showing the

lengths of the other two sides, correct to 2 decimal places.

**INVESTIGATION **

**Cutting rectangles **

Elspeth works in an upholstery business. One of her jobs is cutting out rectangular pieces of foam to make seat cushions. She always has a problem judging whether the cut foam is square.

• Elspeth thinks that the foam in the diagram is square.

Is she right? '---"

### /

I -540mm-+- I'-### "

### ✓

*r:::,<S'*

*�*• Describe a process that Elspeth could use to check whether her foam blocks are square.

]n a builcling €©1�text, 'sqtiare' means, 'at vigl�t angles'.