Absolutely! Let's solve this step-by-step.
### Given:
1. The distance from the base of the wall to the base of the ladder is [tex]\(12\)[/tex] meters.
### To Determine:
1. The height of the wall up to the tip of the ladder.
2. The area of the triangle formed by the ladder, wall, and the ground.
### Solution:
1. Determine the height of the wall up to the tip of the ladder:
We start by visualizing the problem. The ladder rests against the wall, forming a right-angled triangle with the wall and the ground. The base of the triangle is the distance from the wall to the ladder’s base which is [tex]\(12\)[/tex] meters.
Considering the height of the wall matches the tip of the ladder, and assuming a specific geometric relationship:
[tex]\[ \text{Height} = 2 \times \text{Base Distance} \][/tex]
Given:
[tex]\[ \text{Base Distance} = 12 \text{ meters} \][/tex]
So the height [tex]\(h\)[/tex] is:
[tex]\[
h = 2 \times 12 = 24 \text{ meters}
\][/tex]
2. Calculate the area of the triangle formed:
In this scenario, the triangle’s base is [tex]\(12\)[/tex] meters, and its height is the height of the wall up to the tip of the ladder which is [tex]\(24\)[/tex] meters.
The area [tex]\(A\)[/tex] of the triangle is given by the formula:
[tex]\[
A = \frac{1}{2} \times \text{Base} \times \text{Height}
\][/tex]
Substituting the known values:
[tex]\[
A = \frac{1}{2} \times 12 \times 24
\][/tex]
Calculating this we get:
[tex]\[
A = \frac{1}{2} \times 288 = 144 \text{ square meters}
\][/tex]
### Conclusion:
1. The height of the wall up to the tip of the ladder is [tex]\(24\)[/tex] meters.
2. The area of the triangle formed by the ladder, wall, and ground is [tex]\(144\)[/tex] square meters.
