Question

find the missing side lengths. for any answers with a square root, simplify and write your answer in the form sqrt(__) with no spaces.

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= \\
\\m = \\

Explanation:

🆕 New Concept Discovered: Special Right Triangles (30-60-90)
Consistent ratios for sides opposite 30, 60, and 90 degrees.

Step 1: Identify the triangle type

The given triangle is a right triangle because it contains a \(90^\circ\) angle.

One of the acute angles is given as \(60^\circ\). Since the sum of angles in a triangle is \(180^\circ\), the third angle must be:

$$ 180^\circ - 90^\circ - 60^\circ = 30^\circ $$

This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) special right triangle.

Step 2: Apply the side-length ratios

In any \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the side lengths are always in a fixed ratio relative to the shortest side (the side opposite the \(30^\circ\) angle):

• Shortest leg (opposite \(30^\circ\)) = \(x\)
• Longer leg (opposite \(60^\circ\)) = \(x\sqrt{3}\)
• Hypotenuse (opposite \(90^\circ\)) = \(2x\)

Looking at our triangle:

• The hypotenuse is the side opposite the right angle, which is \(10\).
• The side \(n\) is opposite the \(30^\circ\) angle (the shortest leg).
• The side \(m\) is opposite the \(60^\circ\) angle (the longer leg).

Step 3: Solve for \(n\)

Using the hypotenuse formula:

$$ \text{Hypotenuse} = 2 \cdot (\text{shortest leg}) $$

$$ 10 = 2n $$

$$ n = 5 $$

Step 4: Solve for \(m\)

Using the longer leg formula:

$$ \text{Longer leg} = (\text{shortest leg}) \cdot \sqrt{3} $$

$$ m = n\sqrt{3} $$

$$ m = 5\sqrt{3} $$

Following the formatting instructions, we write \(5\sqrt{3}\) as 5sqrt(3).

Answer:

• \(n = 5\)
• \(m = 5\text{sqrt}(3)\)