Review: the whole course

Geometry, grade 10 (5 units) Lesson 26 of 26Practice

Review: the whole course

The vertices of the triangle $\displaystyle ABC$ are $\displaystyle A(0,0)$ , $\displaystyle B(8,0)$ and $\displaystyle C(2,6)$ .
$\displaystyle D$ is the midpoint of $\displaystyle BC$ .

Find the coordinates of $\displaystyle D$ .

\displaystyle x_D

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\displaystyle y_D

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Write the equation of the median $\displaystyle AD$ .

\displaystyle A=(0,0)

\displaystyle B=(8,0)

\displaystyle C=(2,6)

$\displaystyle G$ is the meeting point of the medians, so $\displaystyle {AG}={2GD}$ .
Find the coordinates of $\displaystyle G$ .

\displaystyle x_G

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\displaystyle y_G

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The centre of a circle is on the $\displaystyle y$ -axis, and the circle passes through $\displaystyle A(3,5)$ and $\displaystyle B(5,1)$ .

The centre is

\displaystyle M(0,b)

. Find

\displaystyle b

from

\displaystyle {MA}={MB}

Write the equation of the circle.

\displaystyle A=(3,5)

\displaystyle B=(5,1)

Find the points where the line $\displaystyle {y}={x+2}$ meets the circle.

In the triangle $\displaystyle ABC$ , $\displaystyle D$ , $\displaystyle E$ and $\displaystyle F$ are the midpoints of $\displaystyle AB$ , $\displaystyle AC$ and $\displaystyle BC$ .

Prove: $\displaystyle ADFE$ is a parallelogram.

The medians $\displaystyle BE$ and $\displaystyle CD$ meet at $\displaystyle G$ , and $\displaystyle {BE}={12}$ .

\displaystyle BG

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\displaystyle GE

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In the triangle $\displaystyle ABC$ , $\displaystyle {AB}={12}$ , $\displaystyle {AC}={8}$ and $\displaystyle {BC}={15}$ .
$\displaystyle AD$ bisects the angle $\displaystyle A$ , and $\displaystyle D$ is on $\displaystyle BC$ .
The line through $\displaystyle D$ parallel to $\displaystyle AC$ meets $\displaystyle AB$ at $\displaystyle E$ .

Find

\displaystyle BD

Find

\displaystyle DE

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Prove: $\displaystyle {AE}={ED}$ .

The areas of two similar triangles are $\displaystyle 18$ and $\displaystyle 50$ .

Find the similarity ratio – the large triangle to the small one.

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The perimeter of the small triangle is

\displaystyle 24

. Find the perimeter of the large one.

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The radius of the circle inscribed in the large triangle is

\displaystyle 4

. Find that of the small one.

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Answer and explain.

$\displaystyle A$ and $\displaystyle B$ are two points, and $\displaystyle \ell$ is a line that is not perpendicular to $\displaystyle AB$ . How many points are at equal distances from $\displaystyle A$ and from $\displaystyle B$ , and also at distance $\displaystyle 3$ from $\displaystyle \ell$ ? What locus does each condition give?
In the triangles $\displaystyle ABC$ and $\displaystyle DEF$ , $\displaystyle {AB}={DE}={7}$ , $\displaystyle {AC}={DF}={5}$ and $\displaystyle {\angle C}={\angle F}={60^{\circ}}$ . Are the triangles congruent?
In the same triangles, suppose instead that $\displaystyle {\angle B}={\angle E}={40^{\circ}}$ . Must they be congruent?

$\displaystyle AB$ is a diameter of the circle with centre $\displaystyle O$ , and the line $\displaystyle BT$ is a tangent to the circle at $\displaystyle B$ .
$\displaystyle C$ is on the circle, and the line $\displaystyle AC$ meets the tangent at $\displaystyle E$ .

Prove: $\displaystyle \triangle ABC\sim\triangle AEB$ .

\displaystyle {AC}={4}

and

\displaystyle {CE}={5}

. Find the diameter

\displaystyle AB

. (A hint:

\displaystyle {AB^{2}}={AC\cdot AE}

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Find

\displaystyle BE

, in its simplest form.

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Formulas: geometry, grade 10 (5 units)

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Introduction to analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The midpoint of the segment
$\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The slope of a line through two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line of slope $\displaystyle m$ through the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The explicit equation of a line – $\displaystyle m$ the slope, $\displaystyle b$ where it crosses the $\displaystyle y$ -axis
$\displaystyle y=mx+b$
The general equation of a line
$\displaystyle Ax+By+C=0$
Parallel lines
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$
Perpendicular lines
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
A canonical circle – its centre at the origin, $\displaystyle R$ its radius
$\displaystyle x^{2}+y^{2}=R^{2}$
The circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$

Midsegments and the meeting point of the medians

A triangle’s midsegment – $\displaystyle DE$ is parallel to $\displaystyle BC$
$\displaystyle DE$ $\displaystyle =$ $\displaystyle \frac{1}{2}BC$
A trapezoid’s midsegment – $\displaystyle a$ , $\displaystyle b$ the bases
$\displaystyle EF$ $\displaystyle =$ $\displaystyle \frac{a+b}{2}$
The meeting point of the medians $\displaystyle G$ divides the median $\displaystyle AD$ in the ratio $\displaystyle 2:1$
$\displaystyle AG$ $\displaystyle =$ $\displaystyle 2GD$
Pythagoras’ theorem and its converse: the triangle is right-angled, the right angle opposite $\displaystyle c$
$\displaystyle a^{2}+b^{2}=c^{2}$

Proportion and similarity

A property of a proportion
$\displaystyle \frac{a}{b}=\frac{c}{d}\ \Rightarrow\ \frac{a+b}{b}=\frac{c+d}{d}$
Thales’ theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{DB}$ $\displaystyle =$ $\displaystyle \frac{AE}{EC}$
The extended Thales theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$
The angle bisector theorem – $\displaystyle AD$ bisects the angle $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$
The similarity ratio – corresponding sides
$\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k$
Heights, medians, perimeters, the radii of the circumscribed and of the inscribed circles
$\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{P_1}{P_2}=\frac{R_1}{R_2}=\frac{r_1}{r_2}=k$
The ratio of the areas
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$

The circle

An inscribed angle is half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
An inscribed angle on the diameter $\displaystyle AB$
$\displaystyle \angle ACB=90^{\circ}$
A quadrilateral $\displaystyle ABCD$ inscribed in a circle – opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The angle between the tangent $\displaystyle AT$ and the chord $\displaystyle AB$ equals an inscribed angle on $\displaystyle AB$ from its other side
$\displaystyle \angle BAT$ $\displaystyle =$ $\displaystyle \angle ACB$
Two tangents from the point $\displaystyle P$
$\displaystyle PA=PB$
A quadrilateral $\displaystyle ABCD$ circumscribing a circle
$\displaystyle AB+CD$ $\displaystyle =$ $\displaystyle BC+AD$
The chords $\displaystyle AB$ and $\displaystyle CD$ meet at the point $\displaystyle E$
$\displaystyle AE\cdot EB$ $\displaystyle =$ $\displaystyle CE\cdot ED$
Two secants from the point $\displaystyle P$
$\displaystyle PA\cdot PB$ $\displaystyle =$ $\displaystyle PC\cdot PD$
A tangent $\displaystyle PT$ and a secant $\displaystyle PAB$ from the point $\displaystyle P$
$\displaystyle PT^{2}$ $\displaystyle =$ $\displaystyle PA\cdot PB$