To find the area of the right-angled triangle given its perimeter and hypotenuse, we can follow these steps:
1. Understand the Given Information:
- The perimeter of the triangle is 180 cm.
- The hypotenuse of the triangle is 78 cm.
2. Set Up the Equations:
For a right-angled triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex] (legs), and [tex]\(c\)[/tex] (hypotenuse), we have:
[tex]\[
a + b + c = \text{perimeter}
\][/tex]
Hence,
[tex]\[
a + b + 78 = 180
\][/tex]
3. Solve for [tex]\(a + b\)[/tex]:
Rearrange the equation to find the sum of the other two sides:
[tex]\[
a + b = 180 - 78
\][/tex]
[tex]\[
a + b = 102
\][/tex]
4. Apply the Pythagorean Theorem:
The Pythagorean theorem in a right-angled triangle states:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Given the hypotenuse [tex]\(c = 78\)[/tex], we find:
[tex]\[
78^2 = a^2 + b^2
\][/tex]
[tex]\[
6084 = a^2 + b^2
\][/tex]
5. Form an Auxiliary Equation:
Recall that:
[tex]\[
(a + b)^2 = a^2 + b^2 + 2ab
\][/tex]
Substituting the known values:
[tex]\[
102^2 = 6084 + 2ab
\][/tex]
[tex]\[
10404 = 6084 + 2ab
\][/tex]
6. Solve for [tex]\(ab\)[/tex]:
Rearrange to find [tex]\(ab\)[/tex]:
[tex]\[
10404 - 6084 = 2ab
\][/tex]
[tex]\[
4320 = 2ab
\][/tex]
[tex]\[
ab = 2160
\][/tex]
7. Calculate the Area of the Triangle:
The area of a right-angled triangle is:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
Here, base times height [tex]\(ab\)[/tex] is given by 2160. Therefore:
[tex]\[
\text{Area} = \frac{1}{2} \times 2160
\][/tex]
[tex]\[
\text{Area} = 1080 \text{ square cm}
\][/tex]
So, the area of the right-angled triangle is [tex]\(1080\)[/tex] square cm.
