Problems involving general triangles

Choose between the sine rule, cosine rule and area formula using the congruence tests, and solve two-step problems including bearings and heights.

Choosing a rule

The congruence tests tell you when a triangle is fixed by the information, and which rule to start with:

Given	Congruence test	Start with
Three sides	SSS	cosine rule (an angle)
Two sides and the included angle	SAS	cosine rule (the third side) or area
Two angles and a side	AAS	sine rule
Two sides and a non-included angle	none: may be ambiguous	sine rule, checking both angles

Example 1: bearings●●

Ship $P$ sails $15$ km on a bearing of $040^\circ\text{T}$. Ship $Q$ leaves the same port and sails $20$ km on a bearing of $110^\circ\text{T}$. How far apart are they, to the nearest km?

Included angle $70^\circ$.

The angle at the port between the two paths is 110° − 40°.

$d^2 = 15^2 + 20^2 - 2(15)(20)\cos 70^\circ \approx 419.8$

Two sides and the included angle: cosine rule.

$d \approx 20$ km

Example 2: two steps●●●

From point $A$, the angle of elevation of the top $T$ of a tower is $30^\circ$. From $B$, $20$ m closer to the tower, it is $45^\circ$. Find the height of the tower to the nearest metre.

In $\triangle ABT$: $A = 30^\circ$, $B = 135^\circ$, $T = 15^\circ$ and $AB = 20$.

Work in triangle ABT. The angle at B is 180° − 45° = 135°, so the angle at T is 180° − 30° − 135° = 15°.

$BT = \dfrac{20\sin 30^\circ}{\sin 15^\circ} \approx 38.6$ m

Sine rule for BT, opposite the 30° angle.

$h = BT\sin 45^\circ \approx 27$ m

Now the right-angled triangle at the base: height = BT sin 45°.

Flashcards

SSS given

Cosine rule for an angle.

SAS given

Cosine rule for the third side; area with ½ab sin C.

AAS given

Sine rule.

Two-step problems

Find the shared side first, and don't round it.

Practice

Ship $A$ sails $8$ km on a bearing of $030^\circ\text{T}$ and ship $B$ sails $10$ km on a bearing of $120^\circ\text{T}$ from the same port. How far apart are they, to 1 decimal place?
You know two sides and the angle between them. To find the third side, use:
You know all three angles and no sides. The triangle is:
Two paths leave a point on bearings $040^\circ\text{T}$ and $110^\circ\text{T}$. What angle is between them?
Two paths leave a point on bearings $030^\circ\text{T}$ and $170^\circ\text{T}$. What angle is between them?
Given $a = 7$, $b = 9$ and $A = 40^\circ$, the first step is:
In Example 2, what is the angle of the triangle at the top of the tower, $T$?
A triangle has sides $4$ and $6$ with included angle $90^\circ$. Its area is: