Relations Between Sides and Angles of a Triangle

Find the maximum perimeter of a triangle on a given base  $\text{'}a\text{'}$and having the given vertical angle $\alpha$.

In a triangle $ABC,\mathrm{}\mathrm{}\mathrm{}2ac\sin \frac{1}{2}(A-B+C)=$

In a triangle $ABC, AD$ is the altitude from $A$. Given $b$$>\mathrm{c },\angle C=23°\mathrm{and}AD=\frac{abc}{b^2-c^2},$ then  $\angle B$ is equal to: ( where $a, b$ and $c$ are the lengths of the side opposite to angles $A,B$ and $C$respectively )

Two boats leave the same harbour. One travels on a bearing of $060°$ for $13$ km, the other on a bearing of $135°\text{ for }8$ km. Draw a diagram illustrating their positions and find the angle between their courses and also find distance between two boats. [Use, $\cos 75°=0.92175$]

The distance from Évian-les-Bains to Saint-Gingolph is measured to be $16.9$ kilometers Viewed from Lausanne, there is an angle of $54°$between Évian-les-Bains and Saint-Gingolph. Viewed from Évian-les-Bains, there is an angle of $66°$ between Lausanne and Saint-Gingolph. Évian-les-Bains lies on a bearing of $200°$ from Lausanne.

Find the distance from Lausanne to Évian-les-Bains.

Nyon is $7.0 \mathrm{km}$ from Chens-sur-Léman. Coppet is $6.1 \mathrm{km}$ from Chens-sur-Léman. As viewed from Chens-sur-Léman, there is an angle of $84°$between Nyon and Coppet.

Find the distance from Nyon to Coppet.

The distance from Vevey to Montreux is $6.5 \mathrm{km}$. The distance from Montreux to Saint-Gingolph is $9.6 \mathrm{km}$. The distance from Vevey to Saint-Gingolph is $8.6 \mathrm{km}$ on a bearing of $200°$.

Find the bearing of Saint-Gingolph from Montreux.

The distance from Vevey to Montreux is $6.5 \mathrm{km}$. The distance from Montreux to Saint-Gingolph is $9.6 \mathrm{km}$. The distance from Vevey to Saint-Gingolph is $8.6 \mathrm{km}$ on a bearing of $200°$.

Find the bearing of Montreux from Vevey.

The distance from Vevey to Montreux is $6.5 \mathrm{km}$. The distance from Montreux to Saint-Gingolph is $9.6 \mathrm{km}$. The distance from Vevey to Saint-Gingolph is $8.6 \mathrm{km}$ on a bearing of $200°$.

Find the angle between Montreux and Saint-Gingolph as viewed from Vevey.

I am standing at point $O$, near a major road. Along the road there are markers every $100 \mathrm{m}$. I can see three consecutive markers: $P,Q \text{and} R$. I am $20 \mathrm{m}$ away from marker $P$. The bearing of $P$ from my current position is $340°$ and the bearing of $Q$ from my current position is $030°$.

Find my distance from $R$.

I am standing at point $O$, near a major road. Along the road there are markers every $100 \mathrm{m}$. I can see three consecutive markers: $P,Q \text{and} R$. I am $20 \mathrm{m}$ away from marker $P$. The bearing of $P$ from my current position is $340°$ and the bearing of $Q$ from my current position is $030°$.

Find my distance from $Q$.

I am standing at point $O$, near a major road. Along the road there are markers every $100 \mathrm{m}$. I can see three consecutive markers: $P,Q \text{and} R$. I am $20 \mathrm{m}$ away from marker $P$. The bearing of $P$ from my current position is $340°$ and the bearing of $Q$ from my current position is $030°$.

Find the bearing of $Q$ from $P$.

Peter, Alex and Mary are sea-kayaking. Peter is $430 \mathrm{m}$ from Alex on a bearing of $113°$. Mary is on a bearing of $210°$ and a distance of $310 \mathrm{m}$ from Alex. Find the distance between Peter and Mary.

Two boats leave a harbour: one travels on a bearing of $050°$ for $14 \mathrm{km}$, the other on a bearing of $145°$ for $18 \mathrm{km}$.

Find the distance between the two boats.

From point $P$, a ship sails $5.5 \mathrm{km}$ to point $Q$ on a bearing of $039°$. It then sails $4.1 \mathrm{km}$ to point $R$ on a bearing of $072°$.

Find the distance from $R$ to $P$.

Explain why you cannot use the sine rule to find length $x$ in this triangle.

In triangle $XYZ$, $x=16 \text{cm}$, $\angle Y=44°$ and $\angle Z=72°$. Find the length of side $y$.

Find the value of $x$ for the following triangle.

Find the value of $x$ for the following triangle.