Right-Angled Triangle Properties

We are given a triangle ABC, which is right-angled at vertex B.

• The length of the hypotenuse AC is given as \(\frac{(p+q)}{2}\).
• The length of one side BC is given as \(\frac{(p-q)}{2}\).

Calculating Side AB using Pythagorean Theorem

According to the Pythagorean theorem in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, \(AB^2 + BC^2 = AC^2\).

We can rearrange this to find the length of side AB:

\(AB^2 = AC^2 - BC^2\)

Substitute the given values for AC and BC:

\(AB^2 = \left(\frac{p+q}{2}\right)^2 - \left(\frac{p-q}{2}\right)^2\)

Expand the squares and combine the terms:

\(AB^2 = \frac{(p+q)^2}{4} - \frac{(p-q)^2}{4}\)

\(AB^2 = \frac{1}{4} \left[ (p+q)^2 - (p-q)^2 \right]\)

Using the algebraic identity \((a+b)^2 - (a-b)^2 = 4ab\), where \(a=p\) and \(b=q\):

\(AB^2 = \frac{1}{4} [4pq]\)

\(AB^2 = pq\)

Taking the square root to find the length of AB:

\(AB = \sqrt{pq}\)

Evaluating Statement I: Geometric Mean

Statement I asserts that the value of AB is equal to the geometric mean of p and q.

The geometric mean (GM) of two numbers \(p\) and \(q\) is calculated as \(\sqrt{pq}\).

Our calculation shows that \(AB = \sqrt{pq}\).

Therefore, Statement I is correct.

Evaluating Statement II: Triangle Perimeter

Statement II claims that the perimeter of the triangle is \(p(q+1)\).

The perimeter of triangle ABC is the sum of its sides: AB + BC + AC.

Perimeter = \(\sqrt{pq} + \frac{(p-q)}{2} + \frac{(p+q)}{2}\)

Combine the terms for BC and AC:

Perimeter = \(\sqrt{pq} + \frac{(p-q) + (p+q)}{2}\)

Perimeter = \(\sqrt{pq} + \frac{2p}{2}\)

Perimeter = \(\sqrt{pq} + p\)

Statement II proposes the perimeter is \(p(q+1)\), which expands to \(pq + p\).

Comparing the calculated perimeter (\(\sqrt{pq} + p\)) with the proposed perimeter (\(pq + p\)), we see they are not equal in general.

Therefore, Statement II is incorrect.

Final Conclusion

Based on the analysis, only Statement I is correct.