Practice: analytic geometry

Geometry, grade 10 (5 units) Lesson 6 of 26Practice

Practice: analytic geometry

The equation $\displaystyle {(k-2)x+(k+1)y}={3k}$ is a line for each value of $\displaystyle k$ .
For which $\displaystyle k$ is the line:

through the origin?

parallel to the $\displaystyle x$ -axis? parallel to the $\displaystyle y$ -axis?
Write the line’s equation in each case.

perpendicular to the line

\displaystyle {y}={2x+5}

crossing the

\displaystyle y

-axis at the point

\displaystyle (0,7)

The vertices of the triangle $\displaystyle ABC$ are $\displaystyle A(0,0)$ , $\displaystyle B(8,4)$ and $\displaystyle C(2,6)$ .
$\displaystyle CD$ is the altitude to the side $\displaystyle AB$ .

Find the slope of

\displaystyle AB

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Write the equation of the line $\displaystyle CD$ .

\displaystyle y=\frac{1}{2}x

\displaystyle C=(2,6)

Find the coordinates of $\displaystyle D$ .

Find the lengths of $\displaystyle CD$ and of $\displaystyle AB$ , in their simplest form.

\displaystyle CD

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\displaystyle AB

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Find the area of the triangle

\displaystyle ABC

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The centre of a circle is $\displaystyle M(3,-1)$ , and the circle passes through the point $\displaystyle A(7,2)$ .

Find the radius.

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Write the equation of the circle.

\displaystyle M=(3,-1)

\displaystyle A=(7,2)

The circle cuts the $\displaystyle y$ -axis at the points $\displaystyle P$ and $\displaystyle Q$ .
Find them.

Find the length of the chord

\displaystyle PQ

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$\displaystyle N$ is the midpoint of $\displaystyle PQ$ .
Show that $\displaystyle MN\perp PQ$ .

$\displaystyle ABCD$ is a rhombus.
$\displaystyle A(1,2)$ and $\displaystyle C(7,6)$ are two opposite vertices.
The vertex $\displaystyle B$ is on the $\displaystyle y$ -axis.

Find $\displaystyle M$ , the point where the diagonals meet.

\displaystyle x_M

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\displaystyle y_M

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Write the equation of the diagonal $\displaystyle BD$ .
(A hint: the diagonals of a rhombus are perpendicular and bisect each other.)

\displaystyle A=(1,2)

\displaystyle C=(7,6)

Find the vertices $\displaystyle B$ and $\displaystyle D$ .

Find the length of the rhombus’s side.

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The parabola: $\displaystyle {y}={x^{2}-4x+5}$ ; the line: $\displaystyle {y}={x+1}$ .

Mark the meeting points of the parabola and the line.

\displaystyle y=x^{2}-4x+5

\displaystyle y=x+1

Find the meeting points by solving the system.

Find the distance between the meeting points.

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For which

\displaystyle k

is the line

\displaystyle {y}={2x+k}

a tangent to the parabola?

$\displaystyle A(1,1)$ and $\displaystyle B(5,3)$ are two points.

The point $\displaystyle P(x,y)$ is at equal distances from $\displaystyle A$ and from $\displaystyle B$ .
Simplify $\displaystyle {PA^{2}}={PB^{2}}$ and write the equation of the line on which all such points lie.

\displaystyle A=(1,1)

\displaystyle B=(5,3)

Find the point on the line $\displaystyle {y}={x+2}$ at equal distances from $\displaystyle A$ and from $\displaystyle B$ .

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 $\displaystyle P$
$\displaystyle PT^{2}$ $\displaystyle =$ $\displaystyle PA\cdot PB$