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Geometry & Mensuration Mastery Hub: The Industry Foundation
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Q1Domain Verified
A right-angled triangle has sides measuring $3\text{ cm}$, $4\text{ cm}$, and $5\text{ cm}$. If the semi-perimeter of the triangle is increased by $10\%$, and the inradius is kept constant, what is the percentage increase in the area of the triangle?
$21\%$
$12.5\%$
$20\%$
$10\%$
Q2Domain Verified
The initial area is indeed $6 \text{ cm}^2$. The inradius is $1 \text{ cm}$. The new semi-perimeter is $6.6 \text{ cm}$. The new area is $6.6 \times 1 = 6.6 \text{ cm}^2$. The percentage increase is $(6.6 - 6)/6 \times 100\% = 10\%$. There seems to be a misunderstanding in my initial calculation or interpretation. Let's review the formula and its implications. The question states the inradius is kept constant. This means the shape of the triangle must change to accommodate the increased semi-perimeter while maintaining the same inradius. If $A = sr$ and $r$ is constant, then $A$ is directly proportional to $s$. If $s$ increases by $10\%$, $A$ must also increase by $10\%$. This means my initial calculation of $10\%$ was correct. However, the provided correct answer is B ($21\%$). This suggests a deeper conceptual understanding is required, possibly relating to how changing the semi-perimeter while keeping the inradius constant affects the sides and thus the are
Let's re-evaluate the problem assuming the answer B ($21\%$) is correct and work backward to understand the underlying principle. If the area increased by $21\%$, and $A = sr$, with $r$ constant, then $s$ must have increased by $21\%$. This contradicts the problem statement that the semi-perimeter increased by $10\%$. This indicates a potential flaw in the question's premise or my understanding of how the inradius and semi-perimeter behave under such transformations. Let's assume there's a constraint that is not explicitly stated, or a property that links changes in semi-perimeter and inradius in a way that leads to a $21\%$ area increase. Consider a general triangle with sides $a, b, c$. $s = (a+b+c)/2$. $A = \sqrt{s(s-a)(s-
}$. $r = A/s$. If $s$ increases by $10\%$, $s' = 1.1s$. If $r$ is constant, $A' = s'r = 1.1sr = 1.1A$. This implies a $10\%$ increase in area. Let's consider a different interpretation. Perhaps the question implies a scaling of the triangle. If a triangle is scaled by a factor $k$, its sides become $ka, kb, kc$. The new semi-perimeter $s' = k s$. The new area $A' = k^2 A$. The new inradius $r' = A'/s' = (k^2 A) / (ks) = k (A/s) = kr$. If the semi-perimeter is increased by $10\%$, this means $s' = 1.1s$. This implies a scaling factor $k=1.1$. If $k=1.1$, then the new area would be $A' = (1.1)^2 A = 1.21 A$, which is a $21\%$ increase in area. However, if the triangle is scaled by $k=1.1$, the inradius also scales by $k$, meaning $r' = 1.1r$. This contradicts the condition that the inradius is kept constant. This suggests that the problem is not about simple scaling. It's about a transformation where the semi-perimeter changes, and the inradius remains fixe
This implies the *shape* of the triangle must change significantly. Let's revisit the initial triangle: sides 3, 4, 5. $s=6$, $A=6$, $r=1$. New semi-perimeter $s' = 6 \times 1.1 = 6.6$. If $r$ remains $1$, then the new area $A' = s'r = 6.6 \times 1 = 6.6$. The percentage increase in area is $(6.6 - 6)/6 \times 100\% = 10\%$. There is a fundamental conflict between the provided
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Q3Domain Verified
A solid cone with a height of 24 cm and a base radius of 7 cm is melted and recast into a sphere. What is the radius of the sphere, rounded to two decimal places?
7.00 cm
14.00 cm
10.50 cm
3.50 cm
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This domain protocol is rigorously covered in our 2026 Elite Framework. Every mock reflects direct alignment with the official assessment criteria to eliminate performance gaps.
This domain protocol is rigorously covered in our 2026 Elite Framework. Every mock reflects direct alignment with the official assessment criteria to eliminate performance gaps.
This domain protocol is rigorously covered in our 2026 Elite Framework. Every mock reflects direct alignment with the official assessment criteria to eliminate performance gaps.
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