Practice: similar triangles

Geometry, grade 8 Lesson 20 of 21Practice

Practice: similar triangles

Given: $\displaystyle \triangle ABC\sim\triangle DEF$ .
The lengths: $\displaystyle {AB}={8}$ , $\displaystyle {BC}={10}$ , $\displaystyle {CA}={12}$ , $\displaystyle {DE}={6}$ .

Find $\displaystyle EF$ .

Find $\displaystyle FD$ .

Work out the perimeter of each triangle. What is the ratio of the two perimeters?

The perimeter of

\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

The perimeter of

\displaystyle DEF

/ – 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

In triangle $\displaystyle ABC$ , the angle $\displaystyle C$ is right, and $\displaystyle CD$ is the altitude to the hypotenuse.

a. Prove: $\displaystyle \triangle ACD\sim\triangle ABC$ .

Also given: $\displaystyle {AC}={6}$ and $\displaystyle {AB}={10}$ .
Find $\displaystyle AD$ .

In triangle $\displaystyle ABC$ , the point $\displaystyle D$ is on the side $\displaystyle AB$ , and the point $\displaystyle E$ is on the side $\displaystyle AC$ .
Given: $\displaystyle {\angle ADE}={\angle ACB}$ (in the drawing they have the same mark).

a. Prove: $\displaystyle \triangle ADE\sim\triangle ACB$ .

Also given: $\displaystyle {AD}={4}$ , $\displaystyle {AC}={8}$ , $\displaystyle {AE}={3}$ .
Find $\displaystyle AB$ .

A street lamp is $\displaystyle 4.5$ metres tall.
A boy $\displaystyle 1.5$ metres tall stands $\displaystyle 4$ metres from the lamp post, and the lamp at its top casts his shadow.
How long is the boy’s shadow ( $\displaystyle x$ )? Write a proportion and solve it.
Note: the side of the big triangle on the ground is $\displaystyle 4+x$ .

In triangle $\displaystyle ABC$ , the point $\displaystyle D$ is on the side $\displaystyle AB$ , the point $\displaystyle E$ is on the side $\displaystyle AC$ , and $\displaystyle DE\parallel BC$ .
Given: $\displaystyle {AD}={x}$ , $\displaystyle {DB}={6}$ , $\displaystyle {DE}={4}$ , $\displaystyle {BC}={10}$ .
Find $\displaystyle x$ .

True or false? Explain each answer.

Any two equilateral triangles are similar.

Any two isosceles triangles are similar.

Any two right isosceles triangles are similar.

Two right triangles, each with an angle of $\displaystyle 30^\circ$ , are similar.

The sides of one triangle: $\displaystyle 6$ , $\displaystyle 8$ , $\displaystyle 10$ .
The sides of another triangle: $\displaystyle 9$ , $\displaystyle 12$ , $\displaystyle 15$ .

Are the triangles similar? By which theorem? What is the ratio of similarity?

Are the triangles right-angled? Explain with the converse of Pythagoras’ theorem.

Formulas: geometry, grade 8

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