Question

30-60-90 triangle practice
first triangle: right triangle, hypotenuse 12, 30° angle, legs labeled n (opposite 30°) and m (adjacent to 30°)
second triangle: right triangle, hypotenuse 72, 30° angle, legs labeled a (adjacent to 30°) and b (opposite 30°)
third triangle: right triangle, leg 5 (adjacent to 60°), hypotenuse x, other leg y, 60° angle
fourth triangle: right triangle, leg ( 13sqrt{3} ) (opposite 60°), hypotenuse x, other leg y, 60° angle

Explanation:

Let's solve each 30 - 60 - 90 triangle problem one by one. Recall the properties of a 30 - 60 - 90 triangle: the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest one (let's call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

Top - Left Triangle (30° angle, hypotenuse = 12)

Step 1: Find the side opposite 30° (n)

In a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is half the hypotenuse. So, \(n=\frac{1}{2}\times\) hypotenuse.
The hypotenuse is 12, so \(n = \frac{1}{2}\times12=6\).

Step 2: Find the side opposite 60° (m)

The side opposite \(60^{\circ}\) is \(n\sqrt{3}\). Since \(n = 6\), \(m=6\sqrt{3}\).

Top - Right Triangle (30° angle, hypotenuse = 72)

Step 1: Find the side opposite 30° (b)

The side opposite \(30^{\circ}\) is half the hypotenuse. So, \(b=\frac{1}{2}\times72 = 36\).

Step 2: Find the side opposite 60° (a)

The side opposite \(60^{\circ}\) is \(b\sqrt{3}\). Since \(b = 36\), \(a = 36\sqrt{3}\).

Bottom - Left Triangle (60° angle, side adjacent to 60° = 5)

Step 1: Identify the sides

The side adjacent to \(60^{\circ}\) (length 5) is the side opposite \(30^{\circ}\) (let's confirm: in a right - triangle, the angle of \(60^{\circ}\) means the other non - right angle is \(30^{\circ}\)). So, the side opposite \(30^{\circ}\) (\(a = 5\)), the hypotenuse \(x\) is \(2a\), and the side opposite \(60^{\circ}\) (\(y\)) is \(a\sqrt{3}\).

Step 2: Find the hypotenuse (x)

Since the side opposite \(30^{\circ}\) is 5, the hypotenuse \(x = 2\times5=10\).

Step 3: Find the side opposite 60° (y)

The side opposite \(60^{\circ}\) is \(5\sqrt{3}\).

Bottom - Right Triangle (60° angle, side opposite 60° = \(13\sqrt{3}\))

Step 1: Identify the sides

The side opposite \(60^{\circ}\) is \(13\sqrt{3}\). Let the side opposite \(30^{\circ}\) be \(y\). We know that the side opposite \(60^{\circ}=y\sqrt{3}\).

Step 2: Find the side opposite 30° (y)

If \(y\sqrt{3}=13\sqrt{3}\), then \(y = 13\) (divide both sides by \(\sqrt{3}\)).

Step 3: Find the hypotenuse (x)

The hypotenuse \(x = 2y\). Since \(y = 13\), \(x=2\times13 = 26\).

Final Answers:

• Top - Left: \(n = 6\), \(m = 6\sqrt{3}\)
• Top - Right: \(b = 36\), \(a = 36\sqrt{3}\)
• Bottom - Left: \(x = 10\), \(y = 5\sqrt{3}\)
• Bottom - Right: \(y = 13\), \(x = 26\)

Answer:

Let's solve each 30 - 60 - 90 triangle problem one by one. Recall the properties of a 30 - 60 - 90 triangle: the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest one (let's call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).

Top - Left Triangle (30° angle, hypotenuse = 12)

Step 1: Find the side opposite 30° (n)

In a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is half the hypotenuse. So, \(n=\frac{1}{2}\times\) hypotenuse.
The hypotenuse is 12, so \(n = \frac{1}{2}\times12=6\).

Step 2: Find the side opposite 60° (m)

The side opposite \(60^{\circ}\) is \(n\sqrt{3}\). Since \(n = 6\), \(m=6\sqrt{3}\).

Top - Right Triangle (30° angle, hypotenuse = 72)

Step 1: Find the side opposite 30° (b)

The side opposite \(30^{\circ}\) is half the hypotenuse. So, \(b=\frac{1}{2}\times72 = 36\).

Step 2: Find the side opposite 60° (a)

The side opposite \(60^{\circ}\) is \(b\sqrt{3}\). Since \(b = 36\), \(a = 36\sqrt{3}\).

Bottom - Left Triangle (60° angle, side adjacent to 60° = 5)

Step 1: Identify the sides

The side adjacent to \(60^{\circ}\) (length 5) is the side opposite \(30^{\circ}\) (let's confirm: in a right - triangle, the angle of \(60^{\circ}\) means the other non - right angle is \(30^{\circ}\)). So, the side opposite \(30^{\circ}\) (\(a = 5\)), the hypotenuse \(x\) is \(2a\), and the side opposite \(60^{\circ}\) (\(y\)) is \(a\sqrt{3}\).

Step 2: Find the hypotenuse (x)

Since the side opposite \(30^{\circ}\) is 5, the hypotenuse \(x = 2\times5=10\).

Step 3: Find the side opposite 60° (y)

The side opposite \(60^{\circ}\) is \(5\sqrt{3}\).

Bottom - Right Triangle (60° angle, side opposite 60° = \(13\sqrt{3}\))

Step 1: Identify the sides

The side opposite \(60^{\circ}\) is \(13\sqrt{3}\). Let the side opposite \(30^{\circ}\) be \(y\). We know that the side opposite \(60^{\circ}=y\sqrt{3}\).

Step 2: Find the side opposite 30° (y)

If \(y\sqrt{3}=13\sqrt{3}\), then \(y = 13\) (divide both sides by \(\sqrt{3}\)).

Step 3: Find the hypotenuse (x)

The hypotenuse \(x = 2y\). Since \(y = 13\), \(x=2\times13 = 26\).

Final Answers:

• Top - Left: \(n = 6\), \(m = 6\sqrt{3}\)
• Top - Right: \(b = 36\), \(a = 36\sqrt{3}\)
• Bottom - Left: \(x = 10\), \(y = 5\sqrt{3}\)
• Bottom - Right: \(y = 13\), \(x = 26\)