Practice: the sides of a triangle, the median and the isosceles triangle

Geometry, grade 8 Lesson 17 of 21Practice

Practice: the sides of a triangle, the median and the isosceles triangle

1

Is there a triangle with these sides? Give the reason.

a

7 cm, 8 cm and 16 cm

b

10 cm, 12 cm and 20 cm

c

1.5 cm, 2.5 cm and 4 cm

2

A triangle has two sides: one 3 cm long and one 8 cm long.

a

Between which numbers is the length of the third side?

b

The triangle is isosceles. How long is the third side, and what is the triangle’s perimeter? Give the reasons.

The third side

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/ – 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

3

In triangle $\displaystyle ABC$ , the segment $\displaystyle AD$ is the median to the side $\displaystyle BC$ .
$\displaystyle {BD}={9}$ cm and $\displaystyle {BC}={2x+6}$ cm.
Write an equation, and find $\displaystyle x$ .

4

Find the areas, and give the reasons.

a

In triangle $\displaystyle ABC$ , the segment $\displaystyle AD$ is the median to the side $\displaystyle BC$ , and the area of triangle $\displaystyle ABD$ is 21 cm².
What is the area of triangle $\displaystyle ABC$ ?

/ – 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

In triangle $\displaystyle KLM$ , the segment $\displaystyle LN$ is the median to the side $\displaystyle KM$ .
$\displaystyle {KM}={10}$ cm, and the altitude from $\displaystyle L$ to the side $\displaystyle KM$ is 7 cm.
What is the area of triangle $\displaystyle LNM$ ?

/ – 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

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
The angle $\displaystyle A$ is $\displaystyle 3x$ degrees, and the angle $\displaystyle B$ is $\displaystyle x+20$ degrees.
Write an equation, find $\displaystyle x$ , and find the three angles of the triangle.

\displaystyle \angle A

/ – 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

\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

/ – 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 C

/ – 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 triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
The exterior angle at the vertex $\displaystyle C$ , on the extension of the base $\displaystyle BC$ , is $\displaystyle 115^{\circ}$ .
Find the three angles of the triangle, and give the reasons.

\displaystyle \angle A

/ – 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

\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

/ – 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 ACB

/ – 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

In triangle $\displaystyle ABC$ , $\displaystyle {\angle B}={\angle C}$ .
The sides: $\displaystyle {AB}={3y-2}$ , $\displaystyle {AC}={y+6}$ and $\displaystyle {BC}={7}$ (in cm).
Write an equation, find $\displaystyle y$ , and find the triangle’s perimeter.

The perimeter

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8

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

9

In triangle $\displaystyle ABC$ , $\displaystyle {\angle ABC}={\angle ACB}$ .
$\displaystyle BD$ bisects angle $\displaystyle B$ , and $\displaystyle CE$ bisects angle $\displaystyle C$ .
They meet at the point $\displaystyle O$ .
Prove: $\displaystyle {OB}={OC}$ .

10

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
$\displaystyle D$ is the midpoint of the base $\displaystyle BC$ .
$\displaystyle DE$ is perpendicular to $\displaystyle AB$ , and $\displaystyle DF$ is perpendicular to $\displaystyle AC$ .
Prove: $\displaystyle {DE}={DF}$ .

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
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$