To solve this problem, we will use the properties of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle. This type of triangle has specific properties, one of which is that the lengths of the legs are equal and the length of the hypotenuse is the leg length multiplied by [tex]\( \sqrt{2} \)[/tex].
Let's consider two scenarios involving different leg lengths.
### Scenario 1:
Assume one of the legs of the [tex]\( 45^\circ - 45^\circ - 90^\circ \)[/tex] triangle is 6.5 ft.
1. Given leg length: 6.5 ft
2. Hypotenuse calculation: The hypotenuse of a [tex]\( 45^\circ - 45^\circ - 90^\circ \)[/tex] triangle is given by:
[tex]\[
h = \text{leg} \times \sqrt{2}
\][/tex]
3. Substitute the given leg length:
[tex]\[
h = 6.5 \times \sqrt{2}
\][/tex]
4. Calculate the value:
[tex]\[
h \approx 6.5 \times 1.414 \approx 9.192 \text{ ft}
\][/tex]
Thus, the height [tex]\( h \)[/tex] for a leg length of 6.5 ft is approximately 9.192 ft.
### Scenario 2:
Assume one of the legs of the [tex]\( 45^\circ - 45^\circ - 90^\circ \)[/tex] triangle is 13 ft.
1. Given leg length: 13 ft
2. Hypotenuse calculation: The hypotenuse of a [tex]\( 45^\circ - 45^\circ - 90^\circ \)[/tex] triangle is given by:
[tex]\[
h = \text{leg} \times \sqrt{2}
\][/tex]
3. Substitute the given leg length:
[tex]\[
h = 13 \times \sqrt{2}
\][/tex]
4. Calculate the value:
[tex]\[
h \approx 13 \times 1.414 \approx 18.385 \text{ ft}
\][/tex]
Thus, the height [tex]\( h \)[/tex] for a leg length of 13 ft is approximately 18.385 ft.
### Summary:
For the problem involving a [tex]\( 45^\circ - 45^\circ - 90^\circ \)[/tex] triangle, the height values corresponding to the legs 6.5 ft and 13 ft are as follows:
- For a leg length of 6.5 ft, [tex]\( h \approx 9.192 \)[/tex] ft.
- For a leg length of 13 ft, [tex]\( h \approx 18.385 \)[/tex] ft.
These heights are derived from the properties of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle.
