Practice: the law of sines and areas

Geometry, grade 11 (4 units) Lesson 14 of 20Practice

Practice: the law of sines and areas

1

In the triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={42^{\circ}}$ , $\displaystyle {\angle B}={108^{\circ}}$ and $\displaystyle {AC}={15}$ .
Round to two decimal places.

a

Find

\displaystyle BC

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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b

Find

\displaystyle AB

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

2

In the triangle $\displaystyle ABC$ , $\displaystyle {AB}={9}$ and $\displaystyle {\angle C}={120^{\circ}}$ .
Find the radius of the circumscribed circle. Leave the answer with a square root.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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3

In the triangle $\displaystyle ABC$ , $\displaystyle {BC}={10}$ , $\displaystyle {AC}={7}$ and $\displaystyle {\angle A}={64^{\circ}}$ .
Round to two decimal places.

a

Find

\displaystyle \angle B

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find

\displaystyle AB

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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4

A triangular plot has two sides of $\displaystyle 40$ m and $\displaystyle 55$ m, and the angle between them is $\displaystyle 72^{\circ}$ .
What is the area of the plot, in m²? Round to one decimal place.

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5

The side of a rhombus is $\displaystyle 10$ , and one of its angles is $\displaystyle 60^{\circ}$ .
Find the area of the rhombus. Leave the answer with a square root.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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6

In the triangle $\displaystyle ABC$ , $\displaystyle {AB}={6}$ , $\displaystyle {AC}={12}$ , and the angle $\displaystyle A$ is obtuse.
The area of the triangle is $\displaystyle 18$ .
Find $\displaystyle \angle A$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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7

A regular hexagon is inscribed in a circle of radius $\displaystyle 4$ .
Find the area of the hexagon. Leave the answer with a square root. (A hint: join the centre to every vertex.)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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8

Two observers stand at the points $\displaystyle A$ and $\displaystyle B$ on the shore, $\displaystyle 500$ m apart, and see a ship at the point $\displaystyle S$ .
$\displaystyle {\angle SAB}={62^{\circ}}$ and $\displaystyle {\angle SBA}={71^{\circ}}$ .
Round to one decimal place.

a

How far is the ship from the observer at

\displaystyle A

, in metres?

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b

How far is the ship from the observer at

\displaystyle B

, in metres?

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Formulas: geometry, grade 11 (4 units)

The equation of a circle

The canonical equation – the centre at the origin, the radius $\displaystyle R$
$\displaystyle x^{2}+y^{2}=R^{2}$
The general equation – the centre $\displaystyle (a,b)$ , the radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The distance between two points
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
The midpoint of a segment
$\displaystyle x_M=\frac{x_1+x_2}{2},\ y_M=\frac{y_1+y_2}{2}$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$

Angles and chords in a circle

An inscribed angle – half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
A quadrilateral inscribed in a circle – a pair of opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
A chord $\displaystyle AB$ – $\displaystyle d$ its distance from the centre
$\displaystyle \left(\frac{AB}{2}\right)^{2}+d^{2}$ $\displaystyle =$ $\displaystyle R^{2}$

Tangents

Two tangents from a point $\displaystyle P$ – $\displaystyle A$ , $\displaystyle B$ the points of contact
$\displaystyle PA=PB$
The tangent segment $\displaystyle PA$ – $\displaystyle O$ the centre
$\displaystyle PA^{2}+R^{2}$ $\displaystyle =$ $\displaystyle PO^{2}$

Trigonometry, circumference and area

The law of sines – $\displaystyle R$ the radius of the circle through the triangle’s vertices
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The area of a triangle – $\displaystyle \gamma$ the angle between the sides $\displaystyle a$ and $\displaystyle b$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$
The area of a parallelogram – $\displaystyle a$ , $\displaystyle b$ adjacent sides, $\displaystyle \alpha$ the angle between them
$\displaystyle A$ $\displaystyle =$ $\displaystyle ab\sin\alpha$
The sine of an obtuse angle
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The circumference
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi R$
The area of a disc
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi R^{2}$