1. The diagram shows two mathematically similar triangles, T and U. Two corresponding side lengths are 3 cm and 12 cm. The area of triangle T is 5 cm². Find the area of triangle U.... 1. The diagram shows two mathematically similar triangles, T and U. Two corresponding side lengths are 3 cm and 12 cm. The area of triangle T is 5 cm². Find the area of triangle U. 2. Anna walks 31 km at a speed of 5 km/h. Both values are correct to the nearest whole number. Work out the upper bound of the time taken for Anna's walk.
Understand the Problem
The image presents two math problems. The first problem involves two similar triangles, T and U. Given the side lengths of 3 cm and 12 cm and the area of triangle T (5 cm²), we need to find the area of triangle U. The second problem involves Anna walking 31 km at a speed of 5 km/h. Given that both values are correct to the nearest whole number, we need to calculate the upper bound of the time taken for Anna's walk.
Answer
80 7
Answer for screen readers
80 cm² 7 hours
Steps to Solve
- Find the length scale factor
Since the triangles are similar, the ratio of corresponding sides is constant. The length scale factor is the ratio of the side length of triangle U to the side length of triangle T: $ \text{Length scale factor} = \frac{12}{3} = 4 $
- Find the area scale factor
The area scale factor is the square of the length scale factor: $ \text{Area scale factor} = 4^2 = 16 $
- Find the area of Triangle U
Multiply the area of triangle T by the area scale factor to find the area of triangle U: $ \text{Area of U} = 5 \times 16 = 80 \text{ cm}^2 $
- Determine the upper bound for distance
Since the distance is correct to the nearest whole number, the actual distance could be up to 0.5 km more than 31 km. Therefore, the upper bound for the distance is $31 + 0.5 = 31.5$ km.
- Determine the lower bound for speed
Similarly, since the speed is correct to the nearest whole number, the actual speed could be up to 0.5 km/h less than 5 km/h. Therefore, the lower bound for the speed is $5 - 0.5 = 4.5$ km/h.
- Calculate the upper bound for time
To find the upper bound of the time taken, divide the upper bound of the distance by the lower bound of the speed: $ \text{Upper bound of time} = \frac{31.5}{4.5} = 7 \text{ hours} $
80 cm² 7 hours
More Information
The area of triangle U is $80 \text{ cm}^2$. The upper bound of the time taken for Anna's walk is 7 hours.
Tips
A common mistake is forgetting to square the length scale factor when calculating the area scale factor for similar figures. Another mistake for the second part would be finding the lower bound of the time instead of the upper bound.
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