Similar shapes

Geometry and measures

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Two similar triangles have corresponding sides of $3$ cm and $6$ cm. Find the scale factor from the smaller to the larger.

2 marks
Corresponding sides are $4$ cm and $10$ cm. Find the scale factor from the smaller to the larger.

2 marks
Two similar triangles have a pair of corresponding sides $5$ cm and $10$ cm. A side of $8$ cm on the smaller triangle corresponds to a side on the larger triangle. Find that side.

3 marks
The length scale factor between two similar shapes is $3$. The smaller area is $4$ cm$^{2}$. Find the larger area.

3 marks
Two similar rectangles have a pair of corresponding sides $2$ cm and $6$ cm. The other side of the small rectangle is $5$ cm. Find the other side of the large rectangle and both perimeters.

4 marks
Two similar solids have length scale factor $2$. The smaller volume is $10$ cm$^{3}$. Find the larger volume.

4 marks
Two similar cones have heights $6$ cm and $9$ cm. The smaller base radius is $4$ cm and its curved surface area is $40$ cm$^{2}$. Its volume is $32$ cm$^{3}$. Find the larger radius, curved surface area and volume.

5 marks
In the diagram, $DE$ is parallel to $BC$, $AD = 4$ cm, $DB = 6$ cm and $DE = 6$ cm. Find $BC$. The area of triangle $ABC$ is $100$ cm$^{2}$. Find the area of triangle $ADE$.

Triangles ADE and ABC, with DE parallel to BC. Not to scale.

6 marks

Mark scheme

Answer. $2$
- $\dfrac{6}{3} = 2$
Answer. $\dfrac{5}{2}$
- $\dfrac{10}{4} = \dfrac{5}{2}$
Answer. $16$ cm
- The scale factor is $2$.
- $8 \times 2 = 16$ cm
Answer. $36$ cm$^{2}$
- The area scale factor is $3^{2} = 9$.
- $4 \times 9 = 36$ cm$^{2}$
Answer. Large side $15$ cm. Perimeters $14$ cm and $42$ cm.
- Scale factor $= 3$, so the other side is $15$ cm.
- Small perimeter $= 2(2 + 5) = 14$ cm.
- Large perimeter $= 3 \times 14 = 42$ cm.
Answer. $80$ cm$^{3}$
- The volume scale factor is $2^{3} = 8$.
- $10 \times 8 = 80$ cm$^{3}$
Answer. Radius $6$ cm, area $90$ cm$^{2}$, volume $108$ cm$^{3}$
- Length scale factor $= \dfrac{3}{2}$.
- Radius $= 4 \times \dfrac{3}{2} = 6$ cm.
- Area $= 40 \times \dfrac{9}{4} = 90$ cm$^{2}$.
- Volume $= 32 \times \dfrac{27}{8} = 108$ cm$^{3}$.
Answer. $BC = 15$ cm. Area of $ADE = 16$ cm$^{2}$.
- $AB = 10$ cm, so the scale factor from $ADE$ to $ABC$ is $\dfrac{10}{4} = \dfrac{5}{2}$.
- $BC = 6 \times \dfrac{5}{2} = 15$ cm.
- The area scale factor is $\dfrac{25}{4}$.
- Area of $ADE = 100 \times \dfrac{4}{25} = 16$ cm$^{2}$.