In the diagram above, triangle PQR has a right angle at Q. If PQ>QR, then what is the ratio of the area of triangle PQS to the area of triangle RQS? I. Line segment QS is perpendicular to PR and has a length of 12. II. PQR has a perimeter of 60.
Show Hint
In a right triangle, the ratio of the areas of the two smaller triangles formed by the altitude to the hypotenuse is equal to the ratio of the squares of the corresponding legs of the original right triangle. This is a powerful theorem derived from the similarity of the triangles.
Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
Both statements I and II together are sufficient but neither statement is sufficient alone.
Each statement alone is sufficient to answer the question.
Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.
Hide Solution
Verified By Collegedunia
The Correct Option isC
Solution and Explanation
Step 1: Understanding the Concept:
We are asked to find the ratio of the areas of two smaller triangles, \(\triangle PQS\) and \(\triangle RQS\). These two triangles share a common altitude if we consider PS and RS as their bases. Alternatively, they share a common base QS if we consider the altitudes from P and R to the line containing QS. A simpler approach is to use the fact that they share a common vertex Q and their bases PS and RS are collinear.
The ratio of the areas of two triangles that share the same altitude is equal to the ratio of their bases. From the diagram, if QS is the altitude from Q to PR, then triangles PQS and RQS share the altitude QS. This seems to be a misunderstanding of the area formula. Let's reconsider. The triangles PQS and RQS share a common altitude from vertex Q to the base PR. Let's call the length of this altitude \(h\). This is incorrect.
Let's use the definition of area. The two triangles \(\triangle PQS\) and \(\triangle RQS\) share a common altitude if we take PS and RS as bases. Let \(h\) be the length of the altitude from vertex Q to the line containing PR. The base of \(\triangle PQS\) is PS and the base of \(\triangle RQS\) is RS.
Area(\(\triangle PQS\)) = \(\frac{1}{2} \times PS \times h\)
Area(\(\triangle RQS\)) = \(\frac{1}{2} \times RS \times h\)
The ratio is:
\[ \frac{\text{Area}(\triangle PQS)}{\text{Area}(\triangle RQS)} = \frac{\frac{1}{2} \times PS \times h}{\frac{1}{2} \times RS \times h} = \frac{PS}{RS} \]
So, the question is equivalent to "What is the ratio of PS to RS?". Step 2: Detailed Explanation:
We need to find the ratio PS/RS. In a right-angled triangle PQR (right-angled at Q), where QS is the altitude to the hypotenuse PR, we have the geometric mean theorem and other properties related to similar triangles. Specifically, \(\triangle PQS \sim \triangle QRS \sim \triangle PQR\). From similarity, we have the relationship \( \frac{PQ^2}{QR^2} = \frac{PS}{RS} \). So, finding the ratio of the areas is equivalent to finding the ratio of the squares of the legs of the main triangle, \(\frac{PQ^2}{QR^2}\). Analyze Statement I: "Line segment QS is perpendicular to PR and has a length of 12."
This tells us that QS is the altitude from Q to the hypotenuse, and its length is 12. This information alone does not give us the lengths of PQ or QR, so we cannot find their ratio. Statement I is not sufficient. Analyze Statement II: "PQR has a perimeter of 60."
This gives us \(PQ + QR + PR = 60\). This single equation with three unknowns is not enough to determine the ratio of PQ to QR. Statement II is not sufficient. Analyze Statements I and II Together:
We have a right triangle PQR with altitude QS=12. We also know \(PQ + QR + PR = 60\).
Let \(PQ = a\) and \(QR = b\). Then \(PR = \sqrt{a^2 + b^2}\).
The area of \(\triangle PQR\) can be calculated in two ways:
\[ \text{Area} = \frac{1}{2} \times PQ \times QR = \frac{1}{2} ab \]
\[ \text{Area} = \frac{1}{2} \times PR \times QS = \frac{1}{2} \sqrt{a^2+b^2} \times 12 = 6\sqrt{a^2+b^2} \]
So, \(ab = 12\sqrt{a^2+b^2}\).
We also have the perimeter equation: \(a + b + \sqrt{a^2+b^2} = 60\).
We now have a system of two equations with two variables, \(a\) and \(b\).
From the perimeter equation: \(\sqrt{a^2+b^2} = 60 - a - b\).
Substitute this into the area equation: \(ab = 12(60 - a - b) = 720 - 12a - 12b\).
Also, square the modified perimeter equation: \(a^2+b^2 = (60 - (a+b))^2 = 3600 - 120(a+b) + (a+b)^2 = 3600 - 120a - 120b + a^2 + 2ab + b^2\).
\[ 0 = 3600 - 120a - 120b + 2ab \]
\[ 120(a+b) - 2ab = 3600 \implies 60(a+b) - ab = 1800 \]
We have two equations:
1) \(ab + 12a + 12b = 720\)
2) \(ab - 60a - 60b = -1800\)
This system can be solved for \(a\) and \(b\). For example, subtracting (2) from (1) gives \(72(a+b) = 2520\), so \(a+b = 35\). Then \(ab = 720 - 12(35) = 720 - 420 = 300\).
We can find \(a\) and \(b\) by solving the quadratic equation \(x^2 - 35x + 300 = 0\), which gives \(x=15\) and \(x=20\).
Since \(PQ>QR\), we have \(a=20\) and \(b=15\).
Now we can find the required ratio:
\[ \frac{\text{Area}(\triangle PQS)}{\text{Area}(\triangle RQS)} = \frac{PQ^2}{QR^2} = \frac{20^2}{15^2} = \frac{400}{225} = \frac{16}{9} \]
Since we can find a unique ratio, both statements together are sufficient. Step 3: Final Answer:
Neither statement alone is sufficient, but together they are sufficient. This corresponds to option (C).
