Practice: points, segments and lines

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

Practice: points, segments and lines

The vertices of the quadrilateral $\displaystyle ABCD$ are $\displaystyle A(0,0)$ , $\displaystyle B(1,2)$ , $\displaystyle C(7,3)$ and $\displaystyle D(6,1)$ .

Prove in two ways that $\displaystyle ABCD$ is a parallelogram.
(A hint: the slopes of opposite sides; the midpoints of the diagonals.)

Find the lengths of the sides and of the diagonals.

\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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\displaystyle BC

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\displaystyle AC

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\displaystyle BD

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Find the point where the diagonals meet.

\displaystyle x

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\displaystyle y

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Two of the vertices of the triangle $\displaystyle ABC$ are $\displaystyle B(20,11)$ and $\displaystyle C(8,-5)$ .
The equation of the side $\displaystyle AB$ is $\displaystyle {y}={\frac{2}{5}x+3}$ .
The equation of the side $\displaystyle AC$ is $\displaystyle {y}={6x-53}$ .

Find the coordinates of the vertex $\displaystyle A$ .

$\displaystyle D$ is the midpoint of the side $\displaystyle AC$ , and $\displaystyle E$ the midpoint of the side $\displaystyle AB$ .
Find the coordinates of the points $\displaystyle D$ and $\displaystyle E$ .

Find the length of the segment $\displaystyle DE$ .

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Show that the segment $\displaystyle DE$ is parallel to the side $\displaystyle BC$ and equal to half of it.

$\displaystyle ABCD$ is a rectangle.
Its vertices are $\displaystyle A(-5,-5)$ , $\displaystyle B(-6,-1)$ , $\displaystyle C(2,1)$ and $\displaystyle D(3,-3)$ .

\displaystyle A=(-5,-5)

\displaystyle B=(-6,-1)

\displaystyle C=(2,1)

\displaystyle D=(3,-3)

Find the rectangle’s perimeter and area.

The perimeter

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The area

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Write on the board the equation of the diagonal $\displaystyle AC$ .

Find the point where the diagonals meet.
Show that it is on the diagonal $\displaystyle BD$ as well.

Write the equation of the line through the point $\displaystyle P(-1,3)$ perpendicular to the line $\displaystyle {3x+y}={5}$ .

\displaystyle 3x+y=5

\displaystyle P=(-1,3)

The line through the points $\displaystyle (2,k)$ and $\displaystyle (4,7)$ has the slope $\displaystyle 3$ .
Find $\displaystyle k$ .

The point $\displaystyle P$ is on the $\displaystyle x$ -axis, at equal distances from the points $\displaystyle A(1,3)$ and $\displaystyle B(5,1)$ .
Find the $\displaystyle x$ -coordinate of $\displaystyle P$ .
(A hint: $\displaystyle P(x,0)$ , and $\displaystyle {PA^{2}}={PB^{2}}$ .)

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 $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$