Review

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

Review

The trapezium $\displaystyle ABCD$ is inscribed in a circle, and its bases are $\displaystyle AB$ and $\displaystyle DC$ .
Prove: $\displaystyle {AD}={BC}$ .

A circle’s radius is $\displaystyle 10$ , and the chord $\displaystyle AB$ is $\displaystyle 16$ long.
Find the chord’s distance from the centre of the circle.

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The quadrilateral $\displaystyle ABCD$ is inscribed in a circle.
$\displaystyle {\angle ABC}={105^{\circ}}$ and $\displaystyle {\angle ACB}={40^{\circ}}$ .

Find

\displaystyle \angle ADC

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Find

\displaystyle \angle ADB

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$\displaystyle PA$ and $\displaystyle PB$ are tangents to the circle with centre $\displaystyle O$ , at $\displaystyle A$ and $\displaystyle B$ .
The circle’s radius is $\displaystyle 5$ , and $\displaystyle {PA}={12}$ .

Find

\displaystyle OP

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Find

\displaystyle \angle APB

, to two decimal places.

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In the triangle $\displaystyle ABC$ : $\displaystyle {\angle A}={50^{\circ}}$ , $\displaystyle {\angle B}={70^{\circ}}$ and $\displaystyle {BC}={8}$ .
Round to two decimal places.

Find

\displaystyle AC

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Find the radius of the circle through the triangle’s vertices.

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The sides of a parallelogram are $\displaystyle 6$ and $\displaystyle 10$ , and the angle between them is $\displaystyle 120^{\circ}$ .
Find the parallelogram’s area. Leave the answer with a square root.

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The circle’s centre is $\displaystyle M(2,-3)$ , and the point $\displaystyle P(5,1)$ is on the circle.
Find the circle’s equation.

\displaystyle M=(2,-3)

\displaystyle P=(5,1)

The circle $\displaystyle {x^{2}+y^{2}}={25}$ and the line $\displaystyle {y}={x-1}$ are given.
Find the points where the line cuts the circle.

The point $\displaystyle A(6,8)$ is on the circle $\displaystyle {x^{2}+y^{2}}={100}$ .
Find the equation of the tangent to the circle at $\displaystyle A$ .

\displaystyle x^{2}+y^{2}=100

\displaystyle A=(6,8)

A circle is inscribed in a square of side $\displaystyle 10$ cm.
Round to two decimal places.

Find the area of the disc, in cm².

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Find the circumference of the circle, in cm.

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Formulas: geometry, grade 11 (4 units)

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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
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi R^{2}$