Practice: the isosceles triangle

Geometry, grade 9 Lesson 3 of 21Practice

Practice: the isosceles triangle

1

Which sides of the triangle are equal? Find the lengths you can, and give the reasons.

a

In triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={36^{\circ}}$ , $\displaystyle {\angle B}={72^{\circ}}$ and $\displaystyle {AC}={9}$ .

\displaystyle AB

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b

In triangle $\displaystyle DEF$ , $\displaystyle {\angle D}={60^{\circ}}$ , $\displaystyle {\angle E}={60^{\circ}}$ and $\displaystyle {DE}={7}$ .

\displaystyle EF

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\displaystyle DF

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2

In triangle $\displaystyle ABC$ , the exterior angle at the vertex $\displaystyle C$ is $\displaystyle 110^{\circ}$ , and $\displaystyle {\angle A}={40^{\circ}}$ .
Also $\displaystyle {AB}={5x-3}$ and $\displaystyle {AC}={2x+9}$ .

a

Find angle $\displaystyle B$ and angle $\displaystyle ACB$ .

\displaystyle \angle B

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\displaystyle \angle ACB

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b

Find $\displaystyle x$ and the length of the side $\displaystyle AB$ .

\displaystyle x

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\displaystyle AB

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3

In triangle $\displaystyle ABC$ , the altitude $\displaystyle AD$ bisects angle $\displaystyle A$ .
$\displaystyle {\angle B}={50^{\circ}}$ and $\displaystyle {AB}={8}$ .
Find $\displaystyle AC$ , angle $\displaystyle C$ and angle $\displaystyle BAD$ .

\displaystyle AC

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\displaystyle \angle C

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\displaystyle \angle BAD

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4

In triangle $\displaystyle KLM$ , the median $\displaystyle KN$ is perpendicular to the side $\displaystyle LM$ .
$\displaystyle {KL}={3y+1}$ , $\displaystyle {KM}={13}$ and $\displaystyle {\angle L}={64^{\circ}}$ .
Find $\displaystyle y$ and angle $\displaystyle LKM$ .

\displaystyle y

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\displaystyle \angle LKM

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5

Triangle $\displaystyle ABC$ is drawn on the coordinate system.

a

Find the coordinates of the point $\displaystyle M$ , the midpoint of the side $\displaystyle AB$ .

b

Explain why $\displaystyle CM$ is perpendicular to $\displaystyle AB$ .

c

Explain why triangle $\displaystyle ABC$ is isosceles, and find the length of the leg $\displaystyle AC$ by Pythagoras’ theorem.

\displaystyle AC

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6

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
The point $\displaystyle D$ is on $\displaystyle AB$ and the point $\displaystyle E$ on $\displaystyle AC$ , so that $\displaystyle {BD}={CE}$ .
The segments $\displaystyle BE$ and $\displaystyle CD$ meet at the point $\displaystyle O$ .
Prove: $\displaystyle {OB}={OC}$ .

7

In triangle $\displaystyle ABC$ , the median $\displaystyle AD$ is perpendicular to the side $\displaystyle BC$ .
The point $\displaystyle E$ is on $\displaystyle AD$ .
Prove: $\displaystyle {EB}={EC}$ .

8

Triangles $\displaystyle ABC$ and $\displaystyle DBC$ are isosceles with a common base, $\displaystyle BC$ : $\displaystyle {AB}={AC}$ and $\displaystyle {DB}={DC}$ .
The points $\displaystyle A$ and $\displaystyle D$ are on the two sides of $\displaystyle BC$ , and the segment $\displaystyle AD$ meets $\displaystyle BC$ at the point $\displaystyle M$ .
Prove: $\displaystyle AD\perp BC$ .

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