Geometry
⏱ ~3-min readAceMark Guide
What this topic is really about
In a 30-60-90 triangle the sides are in the ratio 1:√3:2, where the side opposite 30° is the short leg and the hypotenuse is twice that length; given the short leg is 6, the hypotenuse equals 2·6 = 12, so choice B is correct. Choice C (6√3) confuses the long leg (√3 times the short leg) with the hypotenuse, leading to an overestimate.
The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem.. The Pythagorean theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares of the legs: 9²+12²=81+144=225, and √225=15, so choice C is correct.
See the mechanism
The Pythagorean theorem is a fundamental concept in geometry that allows us to calculate the length of the hypotenuse of a right triangle, given the lengths of the legs. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.
An exam-style question, fully explained
A right triangle has legs 9 and 12. Length of the hypotenuse:
- Identify what the question tests: A right triangle has legs 9 and 12..
- The Pythagorean theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares of the legs: 9²+12²=81+144=225, and √225=15, so choice C is correct.
- Choice A (13) would be correct for a 5-12-13 triangle, but here the legs are 9 and 12, so 13 underestimates the true length.
- Why it matters: The Pythagorean theorem is a fundamental concept in geometry that allows us to calculate the length of the hypotenuse of a right triangle, given the lengths of the legs. In this case, the lengths of the legs are 9 and 12, so we can use the theorem to find the length of the hypotenuse. By plugging in the values, we get 9² + 12² = 81 + 144 = 225, and the square root of 225 is 15.
Traps the examiner sets
- Some students may mistakenly apply the Pythagorean theorem with incorrect values or forget to take the square root of the result, leading to an incorrect answer. Others may confuse the Pythagorean theorem with other geometric concepts, such as the properties of similar triangles.
Test your recall
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