This set mixes the five lessons in this module. You will match corresponding sides, use a linear scale factor, link it to area and volume, and decide between similar and congruent. The questions go from easier to harder.
Write your own answer on paper first, then open the worked solution. Mark each one as right, slip or stuck in your mistake log, and use the routing section at the end to choose what to revisit. Unit conversions are part of the practice, so show them.
Questions
1. In triangle ABC, A = 35°, B = 80°, C = 65°. In triangle PQR, P = 80°, Q = 65°, R = 35°. Which side of PQR corresponds to AB?
Show answer
Match the angles: A = 35° matches R, and B = 80° matches P. So AB corresponds to RP.
Check: C = 65° matches Q, so BC corresponds to PQ and CA corresponds to QR.
2. Two triangles are similar. A side of 6 cm in the smaller triangle matches a side of 15 cm in the larger. Another side of the smaller triangle is 8 cm. Find the corresponding side in the larger triangle.
Show answer
Scale factor = 15 ÷ 6 = 2.5. Corresponding side = 8 × 2.5 = 20 cm.
Check: 6 × 2.5 = 15, and 20 ÷ 8 = 2.5.
3. A map has scale 1 : 25 000. Two villages are 4.8 cm apart on the map. How far apart are they in kilometres?
Show answer
Real distance = 4.8 × 25 000 = 120 000 cm. Since 100 000 cm = 1 km, this is 1.2 km.
Check: 1 cm on the map is 25 000 cm = 250 m, and 4.8 × 250 = 1200 m = 1.2 km.
4. A shape has area 9 cm². It is enlarged by scale factor 4. Find the area of the image.
Show answer
Area factor = 4² = 16. New area = 9 × 16 = 144 cm².
Check: 9 × 10 = 90 and 9 × 6 = 54, and 90 + 54 = 144.
5. Two similar shapes have areas 32 cm² and 50 cm². A side of the smaller shape is 12 cm. Find the corresponding side of the larger shape.
Show answer
Area ratio = 50/32 = 25/16. Square root = 5/4 = 1.25. Side = 12 × 1.25 = 15 cm.
Check: 1.25² = 1.5625 and 32 × 1.5625 = 50.
6. On a floor plan with scale 1 : 200, a room measures 5 cm by 3 cm. Find the real floor area in m².
Show answer
Plan area = 5 × 3 = 15 cm². Area factor = 200² = 40 000. Real area = 15 × 40 000 = 600 000 cm². Since 10 000 cm² = 1 m², this is 60 m².
Check by lengths: 5 × 200 = 1000 cm = 10 m, and 3 × 200 = 600 cm = 6 m. Then 10 × 6 = 60 m².
7. Two similar jugs are 10 cm and 15 cm tall. The smaller holds 400 ml. How much does the larger hold?
Show answer
Scale factor = 15 ÷ 10 = 1.5. Volume factor = 1.5³ = 3.375. Capacity = 400 × 3.375 = 1350 ml.
Check: 400 × 3 = 1200 and 400 × 0.375 = 150, and 1200 + 150 = 1350.
8. Two similar solids have volumes 16 cm³ and 54 cm³. The surface area of the smaller solid is 40 cm². Find the surface area of the larger solid.
Show answer
Volume ratio = 54/16 = 27/8. Cube root = 3/2, so the length factor is 1.5. Area factor = 1.5² = 2.25. Surface area = 40 × 2.25 = 90 cm².
Check: 40 × 2 = 80 and 40 × 0.25 = 10, and 80 + 10 = 90.
9. Two solid statues are made of the same material and are similar. Their heights are 20 cm and 50 cm. The smaller has mass 0.8 kg. Find the mass of the larger.
Show answer
Scale factor = 50 ÷ 20 = 2.5. Mass is proportional to volume for the same material, so the factor is 2.5³ = 15.625. Mass = 0.8 × 15.625 = 12.5 kg.
Check: 15.625 × 0.8 = 12.5, and 12.5 ÷ 0.8 = 15.625.
10. In triangle ABC, AB = 8 cm, BC = 11 cm and angle A = 40°. In triangle DEF, DE = 8 cm, EF = 11 cm and angle D = 40°. A student writes “congruent by SAS”. Is this reasoning valid? Explain.
Show answer
No. For SAS, the angle must lie between the two given sides. Angle A lies between AB and AC, not between AB and BC. The angle given is opposite the side BC (11 cm), so this is two sides and a non-included angle. That information does not prove congruence.
If you got these wrong
Use the pattern of your errors to choose what to revisit.
- Question 1 or 10 went wrong: you matched by letter order or misjudged where an angle sits. Go back to identifying corresponding sides and similarity versus congruence.
- Question 2 or 3 went wrong: the scale factor was inverted, or a unit conversion slipped. Revisit using a linear scale factor and try the map scale practice tool.
- Question 4, 5 or 6 went wrong: you multiplied by k instead of k², or forgot to square-root an area ratio. Revisit relating area change to a scale factor.
- Question 7, 8 or 9 went wrong: the wrong power was used, or the cube root was missed. Revisit relating volume change to a scale factor.
Keep any question you missed in the mistake log and retest queue and try it again after a few days.
What to do next
Return to the module overview to re-plan your route, or work through angles and geometric reasoning if corresponding angles still feel shaky. If you want someone to check your working and not only your answers, a teacher in online one-to-one Mathematics tuition can go through it with you.