ABC is a triangle right-angled at B. If $\text{AC} = \frac{(p+q)}{2}$ and $\text{BC} = \frac{(p-q)}{2}$, then which of the following statements is/are correct ?I. The value of AB is equal to the geometric mean of p and q.II. The perimeter of the triangle is $p(q+1)$.Select the answer using the code given below :

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CDS - 2026

ABC is a triangle right-angled at B. If $\text{AC} = \frac{(p+q)}{2}$ and $\text{BC} = \frac{(p-q)}{2}$, then which of the following statements is/are correct ?
I. The value of AB is equal to the geometric mean of p and q.
II. The perimeter of the triangle is $p(q+1)$.
Select the answer using the code given below :

This question was previously asked in

CDS 2 2026 Maths Question Paper (13-Sep-2026)

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I only

II only

Both I and II

Neither I nor II

The correct answer is

I only

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.

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