Question

an isosceles right triangle has leg lengths of 4 centimeters. what is the length of the altitude drawn from the right angle to the hypotenuse?

\\(2\text{ cm}\\)
\\(2\sqrt{2}\text{ cm}\\)
\\(4\text{ cm}\\)
\\(4\sqrt{2}\text{ cm}\\)

Explanation:

Identify given values and target

We are given an isosceles right triangle with leg lengths \(a = b = 4\text{ cm}\). We need to find the length of the altitude \(h\) drawn from the right angle to the hypotenuse.

Calculate the hypotenuse length

Using the Triangle Geometry concept, we apply the Pythagorean theorem to find the hypotenuse \(c\):

$$ c = \sqrt{a^2 + b^2} = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2}\text{ cm} $$

Relate area to find the altitude

The area of the right triangle can be calculated in two ways:

1. Using the two perpendicular legs:

$$ \text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times 4 \times 4 = 8\text{ cm}^2 $$

1. Using the hypotenuse and the altitude to it:

$$ \text{Area} = \frac{1}{2} \times c \times h = \frac{1}{2} \times 4\sqrt{2} \times h $$

Solve for the altitude

Equating the two area expressions:

$$ \frac{1}{2} \times 4\sqrt{2} \times h = 8 $$

$$ 2\sqrt{2} \times h = 8 $$

$$ h = \frac{8}{2\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}\text{ cm} $$

Answer:

• (A) \(2\text{ cm}\)
• (B) \(2\sqrt{2}\text{ cm}\) (Correct answer)
• (C) \(4\text{ cm}\)
• (D) \(4\sqrt{2}\text{ cm}\)