Practice: the kite and the trapezoid

Geometry, grade 9 Lesson 11 of 21Practice

Practice: the kite and the trapezoid

In kite $\displaystyle ABCD$ , $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$ .
Find the other angles of the kite, and give the reasons.

$\displaystyle {\angle BAD}={70^{\circ}}$ and $\displaystyle {\angle BCD}={50^{\circ}}$ .

\displaystyle \angle B

/ – 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 \angle D

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$\displaystyle {\angle ABC}={100^{\circ}}$ and $\displaystyle {\angle BAD}={60^{\circ}}$ .

\displaystyle \angle ADC

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\displaystyle \angle BCD

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In trapezoid $\displaystyle ABCD$ the bases are $\displaystyle AB$ and $\displaystyle DC$ .

$\displaystyle {\angle A}={x}$ and $\displaystyle {\angle D}={2x+30^{\circ}}$ .
Find $\displaystyle x$ and the two angles.

\displaystyle x

/ – 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

\displaystyle \angle A

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\displaystyle \angle D

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$\displaystyle {AB}={9}$ and $\displaystyle {DC}={15}$ , and the area of the trapezoid is $\displaystyle 72$ .
What is its height?

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In kite $\displaystyle ABCD$ , $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$ .
The diagonals meet at the point $\displaystyle M$ .
Also $\displaystyle {BD}={12}$ , $\displaystyle {AM}={8}$ and $\displaystyle {CM}={4.5}$ .

Find the length of $\displaystyle AB$ .

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Find the length of $\displaystyle CB$ .

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Find the perimeter of the kite.

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$\displaystyle ABCD$ is an isosceles trapezoid: $\displaystyle AB\parallel DC$ and $\displaystyle {AD}={BC}$ .

Angle $\displaystyle DAB$ is three times angle $\displaystyle ADC$ .
Find all the angles of the trapezoid.

\displaystyle \angle DAB

/ – 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

\displaystyle \angle ABC

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\displaystyle \angle BCD

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\displaystyle \angle ADC

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The perimeter of the trapezoid is $\displaystyle 40$ , each leg is $\displaystyle 8$ , and the base $\displaystyle DC$ is twice as long as the base $\displaystyle AB$ .
Find the lengths of the bases.

\displaystyle AB

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\displaystyle DC

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

Find the lengths of the four sides, and show that $\displaystyle ABCD$ is a kite.

\displaystyle AB

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\displaystyle BC

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\displaystyle CD

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\displaystyle DA

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The diagonals meet at the point $\displaystyle M$ .
What are the coordinates of $\displaystyle M$ ? Show that $\displaystyle M$ is the midpoint of $\displaystyle BD$ , and that the diagonals are perpendicular.

The slope of a segment: the change in

\displaystyle y

divided by the change in

\displaystyle x

. Parallel segments have the same slope.

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

Find the slope of $\displaystyle KL$ and the slope of $\displaystyle NM$ . What follows?

\displaystyle KL

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\displaystyle NM

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Show that $\displaystyle KLMN$ is a trapezoid. Check $\displaystyle KN$ and $\displaystyle LM$ too.

Is the trapezoid isosceles? Give the reason.

In kite $\displaystyle ABCD$ , $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$ .
The point $\displaystyle E$ is on the main diagonal $\displaystyle AC$ .
Prove: $\displaystyle {EB}={ED}$ .

In trapezoid $\displaystyle ABCD$ the bases are $\displaystyle AB$ and $\displaystyle DC$ , and $\displaystyle {AB}={BC}$ .
Prove that the diagonal $\displaystyle CA$ bisects angle $\displaystyle BCD$ , that is, $\displaystyle {\angle BCA}={\angle ACD}$ .

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
$\displaystyle BD$ bisects angle $\displaystyle B$ , and $\displaystyle CE$ bisects angle $\displaystyle C$ .
The point $\displaystyle D$ is on $\displaystyle AC$ , and $\displaystyle E$ on $\displaystyle AB$ .
Prove: $\displaystyle EBCD$ is an isosceles trapezoid.

Formulas: geometry, grade 9

Open as a page

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
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$