HSPT Math Question of the Day

1. $4\sqrt{2}\text{ cm}$
2. $6\text{ cm}$
3. $8\text{ cm}$ (correct answer)
4. $8\sqrt{2}\text{ cm}$

Explanation: When you encounter an isosceles right triangle problem, remember that this special triangle has two equal legs and specific angle measures: 45°-45°-90°. The key insight is knowing the relationship between the sides. In a 45-45-90 triangle, if each leg has length $x$, then the hypotenuse has length $x\sqrt{2}$. This comes from the Pythagorean theorem: $x^2 + x^2 = \text{hypotenuse}^2$, which simplifies to $2x^2 = \text{hypotenuse}^2$, so hypotenuse = $x\sqrt{2}$. Given that the hypotenuse is $8\sqrt{2}$ cm, you can set up the equation: $x\sqrt{2} = 8\sqrt{2}$. Dividing both sides by $\sqrt{2}$ gives you $x = 8$ cm. Therefore, each leg is 8 cm long. Let's examine why the other answers are incorrect. Choice A ($4\sqrt{2}$ cm) would give a hypotenuse of $4\sqrt{2} \times \sqrt{2} = 4 \times 2 = 8$ cm, which is too small. Choice B (6 cm) would produce a hypotenuse of $6\sqrt{2}$ cm, again too small. Choice D ($8\sqrt{2}$ cm) incorrectly assumes the legs equal the hypotenuse, which violates the fundamental property that the hypotenuse is the longest side in any right triangle. Study tip: Memorize the 45-45-90 triangle ratio: if legs = $x$, then hypotenuse = $x\sqrt{2}$. This pattern appears frequently on standardized tests and will save you time compared to using the Pythagorean theorem from scratch.