Review: the whole course

Geometry, grade 10 (4 units) Lesson 20 of 20Practice

Review: the whole course

The vertices of the quadrilateral $\displaystyle ABCD$ are $\displaystyle A(0,0)$ , $\displaystyle B(8,2)$ , $\displaystyle C(10,8)$ and $\displaystyle D(2,6)$ .
The side $\displaystyle BC$ is extended by its own length, to the point $\displaystyle E(12,14)$ .
The segment $\displaystyle AE$ crosses the side $\displaystyle CD$ at the point $\displaystyle F$ .

What kind of quadrilateral is $\displaystyle ABCD$ ?
Prove it by the slopes.

Find the lengths of the sides $\displaystyle AB$ and $\displaystyle BC$ .
Is $\displaystyle ABCD$ a rhombus?

\displaystyle AB

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\displaystyle BC

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The equation of the line $\displaystyle AE$ is $\displaystyle {y}={\frac{7}{6}x}$ , and the equation of the line $\displaystyle CD$ is $\displaystyle {y}={\frac{1}{4}x+\frac{11}{2}}$ .
Find the coordinates of the point $\displaystyle F$ .

Show that $\displaystyle F$ is the midpoint of the segment $\displaystyle AE$ .
Explain by a midsegment of the triangle $\displaystyle ABE$ why it is.

In the triangle $\displaystyle ABC$ , the point $\displaystyle D$ is on $\displaystyle AB$ and the point $\displaystyle E$ on $\displaystyle AC$ .
$\displaystyle {\angle ADE}={\angle ACB}$ .
Prove: $\displaystyle \triangle ADE\sim\triangle ACB$ .

$\displaystyle {AD}={4}$ , $\displaystyle {AB}={9}$ and $\displaystyle {AE}={6}$ .
Find $\displaystyle AC$ .

In the triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ , the apex angle $\displaystyle {\angle A}={40^{\circ}}$ , and the base $\displaystyle {BC}={8}$ .
Calculate, rounded to two decimal places (only at the end).

\displaystyle AB

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The triangle’s area

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The sides of a parallelogram are $\displaystyle 6$ and $\displaystyle 10$ long, and its area is $\displaystyle 30\sqrt{3}$ .
Find the sine of the parallelogram’s angle.
What are the parallelogram’s angles?

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The acute angle

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The obtuse angle

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$\displaystyle G$ is where the medians of the triangle $\displaystyle ABC$ meet.

The median $\displaystyle AD$ is $\displaystyle 9$ long.
Find $\displaystyle AG$ .

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$\displaystyle A(-2,1)$ , $\displaystyle B(4,3)$ and $\displaystyle C(1,8)$ .
Find the coordinates of $\displaystyle G$ .

\displaystyle x_G

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\displaystyle y_G

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In the triangle $\displaystyle ABC$ , $\displaystyle {AB}={6}$ , $\displaystyle {AC}={4}$ and $\displaystyle {BC}={5}$ .
$\displaystyle AD$ bisects the angle $\displaystyle A$ .
Find $\displaystyle BD$ .

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

Where the perpendicular bisectors meet:

\displaystyle x

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\displaystyle y

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Its distance from the vertices

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A rising line passes through the point $\displaystyle A(1,1)$ , and the angle between it and the $\displaystyle x$ -axis is $\displaystyle 60^{\circ}$ .
Write the line’s equation, exactly.

\displaystyle A=(1,1)

Formulas: geometry, grade 10 (4 units)

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Analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
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
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
A line’s equation in explicit form – $\displaystyle m$ the slope, $\displaystyle b$ where it meets the $\displaystyle y$ -axis
$\displaystyle y=mx+b$
A line’s equation in implicit form
$\displaystyle Ax+By+C=0$
The line of slope $\displaystyle m$ through the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
Parallel lines
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$
Perpendicular lines
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
The slope – $\displaystyle \alpha$ the angle between the line and the positive direction of the $\displaystyle x$ -axis
$\displaystyle m$ $\displaystyle =$ $\displaystyle \tan\alpha$

Similar triangles and midsegments

The similarity ratio – corresponding sides
$\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k$
The ratio of the perimeters
$\displaystyle \frac{P_1}{P_2}$ $\displaystyle =$ $\displaystyle k$
The ratio of the areas
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$
The ratio of corresponding altitudes, medians and angle bisectors
$\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{d_1}{d_2}=k$
A triangle’s midsegment – $\displaystyle DE$ parallel to $\displaystyle BC$
$\displaystyle DE$ $\displaystyle =$ $\displaystyle \frac{1}{2}BC$
A trapezoid’s midsegment – $\displaystyle a$ , $\displaystyle b$ the bases
$\displaystyle EF$ $\displaystyle =$ $\displaystyle \frac{a+b}{2}$

Trigonometry

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The tangent from the sine and the cosine
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
The identity from Pythagoras
$\displaystyle \sin^{2}\alpha+\cos^{2}\alpha$ $\displaystyle =$ $\displaystyle 1$
Angles whose sum is $\displaystyle 90^{\circ}$
$\displaystyle \sin(90^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos(90^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of an obtuse angle
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
A triangle’s area – $\displaystyle \gamma$ the angle between the sides $\displaystyle a$ and $\displaystyle b$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

Exact values

$\displaystyle \sin30^{\circ}=\cos60^{\circ}=\frac{1}{2}$
$\displaystyle \sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin60^{\circ}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan30^{\circ}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{3}}{3}$
$\displaystyle \tan45^{\circ}$ $\displaystyle =$ $\displaystyle 1$
$\displaystyle \tan60^{\circ}$ $\displaystyle =$ $\displaystyle \sqrt{3}$

Special lines in a triangle

Where the medians meet, $\displaystyle G$ divides the median $\displaystyle AD$ in the ratio $\displaystyle 2:1$
$\displaystyle AG$ $\displaystyle =$ $\displaystyle 2GD$
Where the medians meet, in coordinates
$\displaystyle G\left(\frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3}\right)$
The median $\displaystyle AD$ splits the triangle into two triangles of equal area
$\displaystyle S_{ABD}$ $\displaystyle =$ $\displaystyle S_{ACD}$
The angle bisector theorem – $\displaystyle AD$ bisects the angle $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$