1. The problem states that two angles, labeled as $3p$ degrees and $7p$ degrees, form a right angle at their intersection. 2. We know that the sum of angles forming a right angle is $90$ degrees. 3. Therefore, we can write the equation: $$3p + 7p = 90$$ 4. Simplify the left side: $$10p = 90$$ 5. Solve for $p$ by dividing both sides by 10: $$p = \frac{90}{10} = 9$$ 6. So, the value of $p$ is $9$ degrees.