Similar‌ ‌Shapes‌

The bases of the triangles are a pair of corresponding sides.

The heights of the triangles are a pair of corresponding sides.

When writing the ratios the order is very important.

The ratio of the heights is $5:15$ which simplifies to $1:3$

The ratio of the bases is $7:20$ this can be written as $1: \frac{20}{7}$ or $1:2.857...$

The triangles are NOT similar shapes. The ratios for the corresponding lengths are NOT the same.

Pair up the sides that have measurements. Make sure you pair up the side mentioned in the question.

The sides AB and PQ are a pair of corresponding sides.

The sides BC and QR are a pair of corresponding sides.

The ratio of the lengths AB : PQ is $9:27$ which simplifies to $1:3$

This gives a scale factor of enlargement from rectangle ABCD to rectangle PQRS of $3.$

The ratio of the lengths BC : QR is also $1:3$

We can use the scale factor $3$ as a multiplier to find the missing length.

The missing side has been found.  QR $= 12 cm$

Alternatively an equation may be formed and solved:

Pair up the sides that have measurements. Make sure you pair up the side mentioned in the question. Use the angles to help you.

The sides AB and DE are a pair of corresponding sides.

The sides BC and EF are a pair of corresponding sides.

The ratio of the lengths AB : DE is $12.5 : 25$ which simplifies to $1 : 2$

This gives a scale factor of enlargement from triangle ABC to triangle DEF of $2.$

But we want a scale factor from DEF to ABC which will be $\frac{1}{2}$.

The ratio of the lengths BC : EF is also $1:2$

We can use the scale factor $\frac{1}{2}$ as a multiplier to find the missing length.

$18\times\frac{1}{2}=9$

The missing side has been found.  BC $= 9 \; cm$

Alternatively an equation may be formed and solved:

Look for equal angles.

angle ACB = angle DCE as vertically opposite angles are equal

There are a pair of parallel sides – AB and DE.  This means that there are equal angles as there are alternate angles.

Angle ABC = angle CED (alternate angles)

Angle BAC = angle CDE (alternate angles)

It makes the problem much easier if you redraw the triangles side-by-side so that it is easier to pair up the corresponding sides.

Pair up the sides that have measurements. Make sure you pair up the side mentioned in the question. Use the angles to help you.

The sides AC and DC are a pair of corresponding sides.

The sides BC and EC are a pair of corresponding sides.

The ratio of the lengths AC : DC is $4 : 10$ which simplifies to $1 : 2.5$

This gives a scale factor of enlargement from ABC to CDE of $2.5$

The ratio of the lengths BC : EC is also $1 : 2.5$

We can use the scale factor $2.5$ as a multiplier to find the missing length.

$x=5 \times 2.5=12.5$

The value of $x$ has been found, $x=12.5$

Alternatively an equation may be formed and solved:

Look for equal angles.

angle EAB = angle DAC as they are common to both triangles

There are a pair of parallel sides – EB and DC.  This means that there are equal angles as there are corresponding angles.

angle AEB = angle ADC (corresponding angles)

angle ABE = angle ACD (corresponding angles)

It makes the problem much easier if you redraw the triangles side-by-side so that it is easier to pair up the corresponding sides.

Pair up the sides that have measurements.  Make sure you pair up the side mentioned in the question.  Use the angles to help you.

The sides AE and AD are a pair of corresponding sides.

The sides EB and DC are a pair of corresponding sides.

The ratio of the lengths AE : AD is $5 : 8$ which simplifies to $1 : \frac{8}{5}$ or $1 : 1.6$

This gives a scale factor of enlargement from ABE to ACD of $1.6$

The ratio of the lengths EB : DC is also $1 : 1.6$

We can use the scale factor $1.6$ as a multiplier to find the missing length.

$x=7 \times 1.6=11.2$

The value of $x$ has been found, $x=11.2$

Alternatively an equation may be formed and solved:

Use the given information to write a ratio and work out the scale factor.

When writing the ratios the order is very important. 

$\begin{aligned} \text{The ratio of the lengths is} \ \text{length} \; A&: \text{length} \; B\\\\ 5&:10\\\\ \text{which simplifies to} \quad 1&:2 \end{aligned}$

The scale factor of enlargement from shape A to shape B is $2$

As we are finding an area we need to square the ratio of the lengths, and square the scale factor.

$\begin{aligned} \text{The ratio of the lengths is -} \ \text{length} \ A&:\text{length} B\\\\ 1&:2\\\\ \text{The ratio of the areas is -} \ \text{area} \ A&:\text{area} \ B\\\\ 1^2&:2^2\\\\ \text{which simplifies to -}\ \ \ \ \ \ 1&:4 \end{aligned}$

We can use the area scale factor $2^2$ or $4$ as a multiplier to find the missing area.

$60\times 4=240$

The area of shape B is $240 \; cm^2$

Alternatively you can form an equation to solve:

Use the given information to write a ratio and work out the scale factor.

When writing the ratios the order is very important. 

​$\begin{aligned} \text{The ratio of the lengths is} \ \text{length } A&: \text{length } B\\\\ 10&:15\\\\ \text{which simplifies to} \quad 1&:1.5 \end{aligned}$

The scale factor of enlargement from shape A to shape B is $1.5$

As we are finding a volume we need to cube the ratio of the lengths, and cube the scale factor.

$\begin{aligned} \text{The ratio of the lengths is -} \ \text{length} \ A&:\text{length} B\\\\ 1&:1.5\\\\ \text{The ratio of the volumes is -} \ \text{volume} \ A&:\text{volume} \ B\\\\ 1^3&:1.5^3\\\\ \text{which simplifies to -}\ \ \ \ \ \ 1&:3.375 \end{aligned}$

We can use the volume scale factor $1.5^3$ or $3.375$ as a multiplier to find the missing volume.

$400\times 3.375=1350$

The volume of shape B is $1350 \; cm^3$

Alternatively you can form an equation to solve: