New Zealand
Level 7 - NCEA Level 2

# Determination of a Triangle

Lesson

This applet starts with a side of length 12 units, and two other lengths of 7 and 4.

Can you make this into a triangle?

Then explore other lengths. What connection can you see between lengths b and c to length a?  What kind of values do b and c take to make a triangle with base a.

This next applet shows the same relationship between the three sides in a different way.  There are 2 circles whose raidii form 2 sides of the triangle. The base of the triangle is formed by a line segment joining the two centres together. Play with this applet and see if you can confirm the same relationship you discovered above.

## Triangle Inequality Theorem

The triangle inequality theorem states that the sum of two side lengths of a triangle is always greater than the third side.

(if you hadn't already found it, just go back to the applet above and test it out)

The triangle inequality theorem can be used to help us determine if a triangle can exist from the lengths of given sides.

##### Example 1

Can a triangle be formed from $3$3 segments of lengths $2,5$2,5 and $4$4

Think: Check the combinations of sides

Do$2+5=7$2+5=7 which is greater than the third side of length $4$4

$2+4=6$2+4=6, which is greater than the third side of length $5$5

and $5+4=9$5+4=9 which is greater than the third side of length $2$2

So a triangle can be made using the lengths $2,4$2,4 and $5$5

##### Example 2

Can a triangle be formed from $3$3 segments of lengths $6,2$6,2 and $3$3

Think: Check the combinations of sides

Do$6+2=8$6+2=8 which is greater than the third side of length $3$3

$2+3=5$2+3=5, which is NOT greater than the third side of length $6$6

and $3+6=9$3+6=9 which is greater than the third side of length $2$2

So because the second combination did not satisfy the triangle inequality, then a triangle can NOT be made using the lengths $6,2$6,2 and $3$3

### Triangle rules

Any of the triangle rules we know can be used to determine a triangle.

• Sum of angles in a triangle is equal to $180$180 degrees.
• The triangle inequality theorem states that the sum of two side lengths of a triangle is always greater than the third side.
• In right-angled triangles, we have the trigonometric ratios of tangent, sine and cosine $\tan\theta=\frac{opposite}{adjacent},\cos\theta=\frac{adjacent}{hypotenuse},\sin\theta=\frac{opposite}{hypotenuse}$tanθ=oppositeadjacent,cosθ=adjacenthypotenuse,sinθ=oppositehypotenuse
• Cosine rule, $a^2=b^2+c^2-2bc\cos a$a2=b2+c22bccosa
• Sine rule, $\frac{a}{\sin\left(A\right)}=\frac{b}{\cos\left(B\right)}=\frac{c}{\tan\left(C\right)}$asin(A)=bcos(B)=ctan(C)

#### Worked Examples

##### question 1

Consider the adjacent figure:

1. Is this a valid triangle?

Yes

A

No

B

Yes

A

No

B

##### question 2

Given three side lengths $9$9, $4$4, and $5$5:

1. Complete the following statements, using $=$=, $>$>, or $<$<:

$9+4\editable{}5$9+45

$9+5\editable{}4$9+54

$5+4\editable{}9$5+49

2. Do these lengths form a valid triangle?

Yes

A

No

B

Yes

A

No

B

### Outcomes

#### M7-4

Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions

#### 91259

Apply trigonometric relationships in solving problems