Practice: Pythagoras' theorem

Geometry, grade 8 Lesson 6 of 21Practice

Practice: Pythagoras' theorem

1

Find the third side of a right triangle.

a

legs: 12 and 16

/ – 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

hypotenuse: 41, leg: 9

/ – 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

legs: 1.2 and 0.5

/ – 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

d

hypotenuse: 26, leg: 24

/ – 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

2

In triangle $\displaystyle ABC$ , the angle $\displaystyle C$ is a right angle.
$\displaystyle {AC}={7}$ and $\displaystyle {AB}={25}$ .
Find $\displaystyle BC$ , the perimeter of the triangle and its area.

\displaystyle BC

/ – 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

The perimeter

/ – 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

The 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

In the trapezoid $\displaystyle ABCD$ , the angles $\displaystyle A$ and $\displaystyle D$ are right angles.
$\displaystyle {AB}={10}$ , $\displaystyle {DC}={4}$ and $\displaystyle {AD}={8}$ .
Find the length of the side $\displaystyle BC$ and the perimeter of the trapezoid.
Hint: draw an altitude from $\displaystyle C$ to $\displaystyle AB$ .

\displaystyle BC

/ – 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

The perimeter

/ – 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

A ship sailed 12 km north, and then 5 km east.
How far is it now from where it started?

/ – 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

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

/ – 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

/ – 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

The sides of a triangle are 15, 20 and 25 long.
Is the triangle right-angled? Explain.
If it is, find its area.

The 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

7

The side of a square is 5 cm.
Find the length of its diagonal.
Between which two whole numbers is it?

/ – 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

8

The diagonal of a rectangle is 17 cm, and one of its sides is 8 cm.
Find the other side and the area of the rectangle.

The other side

/ – 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

The 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

9

A ladder 5 m long leans against a wall, its foot 1.4 m from the wall.
The foot of the ladder slipped, and it is now 3 m from the wall.
How many metres did the top of the ladder go down?

The height at first

/ – 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

The height after it slipped

/ – 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

How far the top went down

/ – 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

Formulas: geometry, grade 8

Angles between parallel lines

Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
Between parallel lines: corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Between parallel lines: co-interior angles
$\displaystyle \gamma+\delta=180^{\circ}$

The angles of a triangle

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
An exterior angle $\displaystyle \delta$ equals the sum of the two interior angles not next to it
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$

Pythagoras' theorem

In a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
Finding a leg – from the hypotenuse and the other leg
$\displaystyle a^2$ $\displaystyle =$ $\displaystyle c^2-b^2$

The sides of a triangle

The triangle inequality – two sides together are longer than the third
$\displaystyle a+b>c$
The third side – between the difference of the other two and their sum
$\displaystyle a-b<c<a+b$ $\displaystyle a\ge b$

Similar triangles

In similar triangles $\displaystyle \triangle ABC\sim\triangle DEF$ the corresponding sides are in one ratio – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{DE}{AB}=\frac{EF}{BC}=\frac{FD}{CA}=k$
The ratio of the perimeters – the perimeter grows $\displaystyle k$ times too
$\displaystyle P_{DEF}$ $\displaystyle =$ $\displaystyle k\cdot P_{ABC}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$