Question

45-45-90 triangles practice
(first triangle: right - angled, one angle 45°, one leg √2, other leg y, hypotenuse x)
(second triangle: right - angled, one angle 45°, one leg 8, other leg n, hypotenuse m)
(third triangle: right - angled, one angle 45°, hypotenuse 9√2, legs a and b)
(fourth triangle: right - angled, one angle 45°, one leg 9, other leg y, hypotenuse x)

Explanation:

Let's solve each 45-45-90 triangle problem. Recall that in a 45-45-90 triangle, the legs are equal, and the hypotenuse \( h = \text{leg} \times \sqrt{2} \), or \( \text{leg} = \frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2} \).

Top - Left Triangle (with leg \( \sqrt{2} \), find \( y \) and \( x \))

Step 1: Identify triangle type

It's a 45-45-90 triangle, so legs are equal. One leg is \( \sqrt{2} \), so \( y = \sqrt{2} \) (other leg).

Step 2: Find hypotenuse \( x \)

Hypotenuse \( x = \text{leg} \times \sqrt{2} = \sqrt{2} \times \sqrt{2} = 2 \).

Top - Right Triangle (leg = 8, find \( n \) and \( m \))

Step 1: Find leg \( n \)

In 45-45-90, legs are equal. One leg is 8, so \( n = 8 \).

Step 2: Find hypotenuse \( m \)

Hypotenuse \( m = 8 \times \sqrt{2} = 8\sqrt{2} \).

Bottom - Left Triangle (hypotenuse \( 9\sqrt{2} \), find \( a \) and \( b \))

Step 1: Find legs \( a \) and \( b \)

Leg \( = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{9\sqrt{2}}{\sqrt{2}} = 9 \).
So \( a = b = 9 \) (legs are equal).

Bottom - Right Triangle (leg = 9, find \( y \) and \( x \))

Step 1: Find leg \( y \)

Legs are equal, so \( y = 9 \).

Step 2: Find hypotenuse \( x \)

Hypotenuse \( x = 9 \times \sqrt{2} = 9\sqrt{2} \).

Final Answers:

• Top - Left: \( y = \boldsymbol{\sqrt{2}} \), \( x = \boldsymbol{2} \)
• Top - Right: \( n = \boldsymbol{8} \), \( m = \boldsymbol{8\sqrt{2}} \)
• Bottom - Left: \( a = \boldsymbol{9} \), \( b = \boldsymbol{9} \)
• Bottom - Right: \( y = \boldsymbol{9} \), \( x = \boldsymbol{9\sqrt{2}} \)

Answer:

Let's solve each 45-45-90 triangle problem. Recall that in a 45-45-90 triangle, the legs are equal, and the hypotenuse \( h = \text{leg} \times \sqrt{2} \), or \( \text{leg} = \frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2} \).

Top - Left Triangle (with leg \( \sqrt{2} \), find \( y \) and \( x \))

Step 1: Identify triangle type

It's a 45-45-90 triangle, so legs are equal. One leg is \( \sqrt{2} \), so \( y = \sqrt{2} \) (other leg).

Step 2: Find hypotenuse \( x \)

Hypotenuse \( x = \text{leg} \times \sqrt{2} = \sqrt{2} \times \sqrt{2} = 2 \).

Top - Right Triangle (leg = 8, find \( n \) and \( m \))

Step 1: Find leg \( n \)

In 45-45-90, legs are equal. One leg is 8, so \( n = 8 \).

Step 2: Find hypotenuse \( m \)

Hypotenuse \( m = 8 \times \sqrt{2} = 8\sqrt{2} \).

Bottom - Left Triangle (hypotenuse \( 9\sqrt{2} \), find \( a \) and \( b \))

Step 1: Find legs \( a \) and \( b \)

Leg \( = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{9\sqrt{2}}{\sqrt{2}} = 9 \).
So \( a = b = 9 \) (legs are equal).

Bottom - Right Triangle (leg = 9, find \( y \) and \( x \))

Step 1: Find leg \( y \)

Legs are equal, so \( y = 9 \).

Step 2: Find hypotenuse \( x \)

Hypotenuse \( x = 9 \times \sqrt{2} = 9\sqrt{2} \).

Final Answers:

• Top - Left: \( y = \boldsymbol{\sqrt{2}} \), \( x = \boldsymbol{2} \)
• Top - Right: \( n = \boldsymbol{8} \), \( m = \boldsymbol{8\sqrt{2}} \)
• Bottom - Left: \( a = \boldsymbol{9} \), \( b = \boldsymbol{9} \)
• Bottom - Right: \( y = \boldsymbol{9} \), \( x = \boldsymbol{9\sqrt{2}} \)