Question

the triangle with side lengths \\(\sqrt{3}, 2, \sqrt{7}\\) is dropdown.

Explanation:

⚡ Using what you learned: the pythagorean theorem and its converse

Step 1: Identify the side lengths and the longest side

Let the side lengths be \(a\), \(b\), and \(c\).

Approximate the values to find the longest side (\(c\)):

$$ a = \sqrt{3} \approx 1.73 $$

$$ b = 2 = \sqrt{4} $$

$$ c = \sqrt{7} \approx 2.65 $$

Since \(\sqrt{7}\) is the largest value, the longest side is \(c = \sqrt{7}\).

Step 2: Compare \(a^2 + b^2\) and \(c^2\)

Calculate the squares of the sides:

$$ a^2 = (\sqrt{3})^2 = 3 $$

$$ b^2 = 2^2 = 4 $$

$$ c^2 = (\sqrt{7})^2 = 7 $$

Sum the squares of the two shorter sides:

$$ a^2 + b^2 = 3 + 4 = 7 $$

Compare this sum to the square of the longest side:

$$ a^2 + b^2 = c^2 \quad (7 = 7) $$

Step 3: Apply the Converse of the Pythagorean Theorem

Since \(a^2 + b^2 = c^2\), the triangle satisfies the Pythagorean theorem. Therefore, it is a right triangle.

Answer:

a right triangle