Practice: lines, planes, prisms and pyramids

Space geometry and vectors, grade 12 (4 units) Lesson 6 of 20Practice

Practice: lines, planes, prisms and pyramids

1

In the box $\displaystyle ABCDA'B'C'D'$ :

a

Write a line and a point that determine the base plane $\displaystyle ABCD$ .

b

The line $\displaystyle BB'$ is the line of intersection of which two planes?

c

What is the position of the planes $\displaystyle ABCD$ and $\displaystyle B'C'CB$ ? If they intersect, write the line of intersection.

d

What is the position of the line $\displaystyle A'D$ and the base plane $\displaystyle ABCD$ ?

e

What is the position of the lines $\displaystyle AC$ and $\displaystyle B'D'$ ?

2

The base of a right prism is the isosceles triangle $\displaystyle ABC$ : $\displaystyle {AB}={AC}={5}$ , $\displaystyle {BC}={6}$ .
The prism is 9 high.

a

The base’s area

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The lateral area

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The surface area

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

3

The base of the right pyramid $\displaystyle SABCD$ is a square of side 10.
Each side edge is 13 long.

a

The height

\displaystyle SM

of the face

\displaystyle SBC

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The surface area

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The pyramid’s height

\displaystyle SO

, exactly

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

4

The edge of the cube $\displaystyle ABCDA'B'C'D'$ is 6 long.

a

The cube’s diagonal

\displaystyle AC'

, exactly

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The angle between

\displaystyle AC'

and the base – the angle

\displaystyle C'AC

, to two decimal places

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

5

The base of the pyramid $\displaystyle SABCD$ is a rectangle: $\displaystyle {AB}={8}$ , $\displaystyle {BC}={6}$ .
The edge $\displaystyle SA$ is perpendicular to the base plane, and $\displaystyle {SA}={6}$ .

a

Why is the triangle $\displaystyle SAC$ right-angled? Explain.

b

\displaystyle SC

, exactly

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The angle

\displaystyle SCA

, to two decimal places

/ – 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

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

6

In the triangular pyramid $\displaystyle SABC$ , which edges are skew to the edge $\displaystyle SA$ ? Explain.

Formulas: space geometry and vectors, grade 12 (4 units)

The right-angled triangle and a triangle's area

Pythagoras’ theorem – $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The cosine of an obtuse angle
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
A triangle’s area – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

The prism

The surface area – the lateral surface and the two bases; $\displaystyle B$ the base’s area
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+2B$
The lateral area of a right prism – $\displaystyle P$ the base’s perimeter, $\displaystyle h$ the height
$\displaystyle M$ $\displaystyle =$ $\displaystyle P\cdot h$
The volume of a prism – $\displaystyle h$ the height, the distance between the bases
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$

The pyramid

The surface area – the lateral surface and the base
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+B$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$

Vectors, geometrically

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
The difference of two vectors leaving the same point
$\displaystyle \overrightarrow{AD}-\overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{BD}$
Collinear vectors
$\displaystyle \vec{v}=t\cdot\vec{u}$
A linear combination of $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$
$\displaystyle \vec{w}=s\vec{u}+t\vec{v}$

Vectors in coordinates

The vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
A vector’s length
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{x^2+y^2+z^2}$
The midpoint of the segment $\displaystyle AB$
$\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)$

The scalar product

The definition – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle |\vec{u}|\cdot|\vec{v}|\cdot\cos\alpha$
In coordinates
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle x_1x_2+y_1y_2+z_1z_2$
The angle between two vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{\vec{u}\cdot\vec{v}}{|\vec{u}|\cdot|\vec{v}|}$
A vector’s length by the scalar product
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{\vec{u}^2}$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})^2$ $\displaystyle =$ $\displaystyle \vec{u}^2+2\vec{u}\cdot\vec{v}+\vec{v}^2$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})\cdot(\vec{u}-\vec{v})$ $\displaystyle =$ $\displaystyle \vec{u}^2-\vec{v}^2$
Perpendicular vectors
$\displaystyle \vec{u}\perp\vec{v}\iff\vec{u}\cdot\vec{v}=0$