Practice: the rectangle, the rhombus and the square

Geometry, grade 9 Lesson 18 of 21Practice

Practice: the rectangle, the rhombus and the square

1

True or false? Give the reason. If a statement is false, describe a quadrilateral where it does not hold.

a

Every rectangle is a parallelogram.

b

Every rhombus is a square.

c

A quadrilateral whose diagonals are equal is a rectangle.

d

A quadrilateral whose diagonals bisect each other and are equal is a rectangle.

e

A quadrilateral whose diagonals are perpendicular is a rhombus.

f

A parallelogram with a right angle and two equal adjacent sides is a square.

2

$\displaystyle ABCD$ is a rectangle, and its diagonals meet at the point $\displaystyle O$ .
$\displaystyle {AC}={16}$ and $\displaystyle {\angle AOB}={120^{\circ}}$ .

a

Find the angle $\displaystyle OAB$ .

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b

Find the length of the side $\displaystyle BC$ .

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c

What kind of triangle is $\displaystyle BOC$ ? Find its perimeter.

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3

$\displaystyle ABCD$ is a rhombus, and its diagonals meet at the point $\displaystyle O$ .
The perimeter of the rhombus is $\displaystyle 36$ , and $\displaystyle {\angle ABC}={120^{\circ}}$ .

a

Find the length of the rhombus’s side.

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b

Find the angle $\displaystyle BAD$ and the length of the diagonal $\displaystyle BD$ .

\displaystyle \angle BAD

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\displaystyle BD

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c

Find the angle $\displaystyle ACB$ .

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4

The quadrilateral $\displaystyle ABCD$ is drawn in the coordinate system.

a

Work out the lengths of the sides of $\displaystyle ABCD$ .

\displaystyle AB

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\displaystyle BC

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\displaystyle CD

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\displaystyle DA

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b

What kind of quadrilateral is $\displaystyle ABCD$ ? Give the reason.

c

Work out the lengths of the diagonals $\displaystyle AC$ and $\displaystyle BD$ . Is $\displaystyle ABCD$ a square? Give the reason.

\displaystyle AC

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\displaystyle BD

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5

$\displaystyle ABCD$ is a rectangle, and the point $\displaystyle M$ is the midpoint of the side $\displaystyle BC$ .
Prove: $\displaystyle {\angle MAD}={\angle MDA}$ .

6

The diagonals of the quadrilateral $\displaystyle ABCD$ meet at the point $\displaystyle O$ , and $\displaystyle {OA}={OB}={OC}={OD}$ .
Prove: $\displaystyle ABCD$ is a rectangle.

7

$\displaystyle ABCD$ is a rhombus.
The point $\displaystyle E$ is on the side $\displaystyle AB$ , and the point $\displaystyle F$ on the side $\displaystyle AD$ , so that $\displaystyle {AE}={AF}$ .
Prove: $\displaystyle {CE}={CF}$ .

8

$\displaystyle ABCD$ is a parallelogram.
The diagonal $\displaystyle AC$ bisects the angle $\displaystyle A$ , and the diagonals are equal: $\displaystyle {AC}={BD}$ .
Prove: $\displaystyle ABCD$ is a square.

9

$\displaystyle ABCD$ is a square.
The point $\displaystyle K$ is on the side $\displaystyle AB$ , the point $\displaystyle L$ on $\displaystyle BC$ , the point $\displaystyle M$ on $\displaystyle CD$ and the point $\displaystyle N$ on $\displaystyle DA$ , so that $\displaystyle {AK}={BL}={CM}={DN}$ .
Prove: $\displaystyle KLMN$ is a square.

Formulas: geometry, grade 9

Angles and sides in a triangle

The sum of 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
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
The larger side faces the larger angle, and the other way round
$\displaystyle a>b\ \Leftrightarrow\ \alpha>\beta$

The right triangle

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^{2}+b^{2}$ $\displaystyle =$ $\displaystyle c^{2}$
The leg $\displaystyle a$ facing a $\displaystyle 30^{\circ}$ angle is half the hypotenuse $\displaystyle c$
$\displaystyle a$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$
The median $\displaystyle m_c$ to the hypotenuse is half the hypotenuse $\displaystyle c$
$\displaystyle m_c$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$

Angles between parallel lines

Corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Co-interior angles
$\displaystyle \alpha+\beta=180^{\circ}$

Angles in a quadrilateral

The sum of the angles of a quadrilateral
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$
In a trapezoid: the two angles beside a leg
$\displaystyle \alpha+\beta=180^{\circ}$

Areas

The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a rectangle
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^{2}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a rhombus or a kite – $\displaystyle d_1$ , $\displaystyle d_2$ the diagonals
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{d_1\cdot d_2}{2}$
The ratio of the areas of similar triangles – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$