Certainly! Let's solve the problem step-by-step.
Given:
- A triangle with angles [tex]\( 45^\circ \)[/tex], [tex]\( 45^\circ \)[/tex], and [tex]\( 90^\circ \)[/tex].
- This is a right-angled isosceles triangle, which means the two legs are equal in length.
### Step 1: Define the sides of the triangle
Let each leg of the triangle be of length [tex]\( a \)[/tex].
### Step 2: Apply the Pythagorean Theorem
In a right-angled triangle, the Pythagorean Theorem states that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Since this triangle is isosceles ([tex]\( 45^\circ \)[/tex], [tex]\( 45^\circ \)[/tex]), the lengths of the two legs ([tex]\( a \)[/tex] and [tex]\( b \)[/tex]) are equal:
[tex]\[ a = b \][/tex]
Substitute [tex]\( a \)[/tex] in place of [tex]\( b \)[/tex]:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
[tex]\[ 2a^2 = c^2 \][/tex]
### Step 3: Solve for [tex]\( c \)[/tex] (the hypotenuse)
To find the hypotenuse [tex]\( c \)[/tex], take the square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = a\sqrt{2} \][/tex]
Thus, we have shown that the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times the length of each leg [tex]\( a \)[/tex].
### Verification:
Using an example where [tex]\( a = 1 \)[/tex] (for simplicity):
- One leg [tex]\( a = 1 \)[/tex]
- Hypotenuse [tex]\( c = 1 \cdot \sqrt{2} = \sqrt{2} \approx 1.4142135623730951 \)[/tex]
#### Calculation:
By substituting [tex]\( a = 1 \)[/tex] into the Pythagorean theorem:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
[tex]\[ 1^2 + 1^2 = c^2 \][/tex]
[tex]\[ 1 + 1 = c^2 \][/tex]
[tex]\[ 2 = c^2 \][/tex]
[tex]\[ c = \sqrt{2} \approx 1.4142135623730951 \][/tex]
Hence, we have demonstrated through both general proof and a specific example that the hypotenuse of a [tex]\( 45^\circ \)[/tex], [tex]\( 45^\circ \)[/tex], [tex]\( 90^\circ \)[/tex] triangle is indeed [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
