One triangle, three tools
Choosing between the Pythagorean theorem, a trigonometric ratio and an area formula on the same figure.
Scenario
A right triangle has its right angle at C. The side AC measures 9 centimetres and the side BC measures 12 centimetres.
A second triangle is similar to it, with its shortest side measuring 15 centimetres.
Answers in order, explanations withheld, and a report on where the chain broke.
- 7.1Right trianglesFoundation
What is the length of AB?
- A15 cm
- B21 cm
- C3 cm
- D10.5 cm
Show the answer and reasoningHide the reasoning
15 cm
AB is the hypotenuse, opposite the right angle at C. So 9^2 + 12^2 = 81 + 144 = 225, and AB = 15 cm. Adding the legs gives 21, which is the wrong operation on the right numbers.
- 7.2Trigonometric ratiosStandard
Builds on: The hypotenuse of 15 cm from question 1, without which no ratio involving it can be written.
What is sin A?
- A12/15
- B9/15
- C12/9
- D9/12
Show the answer and reasoningHide the reasoning
12/15
Sine is opposite over hypotenuse. The side opposite angle A is BC, which is 12, and the hypotenuse is 15, so sin A = 12/15. Using 9 as the opposite side confuses A with B.
- 7.3AreaStandard
What is the area of the first triangle?
- A54 square cm
- B108 square cm
- C67.5 square cm
- D90 square cm
Show the answer and reasoningHide the reasoning
54 square cm
In a right triangle the two legs are a base and its perpendicular height, so the area is (1/2)(9)(12) = 54. Using the hypotenuse as a side gives the wrong figure; it is not perpendicular to either leg.
- 7.4Similar trianglesChallenge
Builds on: The area of 54 from question 3, and the fact from question 1 that 9 is the shortest side rather than 12.
What is the area of the second, similar triangle?
- A150 square cm
- B90 square cm
- C54 square cm
- D270 square cm
Show the answer and reasoningHide the reasoning
150 square cm
The shortest side goes from 9 to 15, a scale factor of 15/9 = 5/3. Area scales with the square of that factor: 54 x (5/3)^2 = 54 x 25/9 = 150. Multiplying the area by 5/3 instead gives 90, which is the trap.