Question:
Square \(ABCD\) has midpoints \(E, F, G, H\) of sides \(AB, BC, CD, DA\) respectively. Let \(L\) be the line through \(F\) and \(H\). Points \(P, Q\) are on \(L\) inside \(ABCD\) such that \(\angle APD = \angle BQC = 120^\circ\). What is the ratio of area of \(ABQCDP\) to the remaining area?
Square \(ABCD\) has midpoints \(E, F, G, H\) of sides \(AB, BC, CD, DA\) respectively. Let \(L\) be the line through \(F\) and \(H\). Points \(P, Q\) are on \(L\) inside \(ABCD\) such that \(\angle APD = \angle BQC = 120^\circ\). What is the ratio of area of \(ABQCDP\) to the remaining area?
Show Hint
For complex geometry ratio problems, coordinate geometry often simplifies calculation.
Updated On: Mar 20, 2026
- \(\frac{4\sqrt{2}}{3}\)
- \(2 + \sqrt{3}\)
- \(\frac{10 - 3\sqrt{3}}{9}\)
- \(1 + \frac1{\sqrt{3}}\)
- \(2\sqrt{3} - 1\)
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
By coordinate geometry, place square of side \(2\), locate midpoints, find intersections.
Calculate polygon \(ABQCDP\) area and compare to remainder.
The ratio simplifies to \(\frac{10 - 3\sqrt{3}}{9}\).
Was this answer helpful?
0
0
Top CAT Quantitative Aptitude Questions
- The set of all real values of $x$ for which $(x^2 - |x+9| + x)>0$ is:
- CAT - 2025
- Quantitative Aptitude
- Number System
- For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is
- CAT - 2025
- Quantitative Aptitude
- Number System
- The rate of water flow through three pipes A, B and C are in the ratio 4 : 9 : 36. An empty tank can be filled up completely by pipe A in 15 hours. If all the three pipes are used simultaneously to fill up this empty tank, the time, in minutes, required to fill up the entire tank completely is nearest to
- CAT - 2025
- Quantitative Aptitude
- Time and Work
- The ratio of expenditures of Lakshmi and Meenakshi is $2 : 3$, and the ratio of income of Lakshmi to expenditure of Meenakshi is $6 : 7$. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is $4 : 9$, then the ratio of their incomes is:
- The average salary of 5 managers and 25 engineers in a company is 60000 rupees. If each of the managers received 20% salary increase while the salary of the engineers remained unchanged, the average salary of all 30 employees would have increased by 5%. The average salary, in rupees, of the engineers is
View More Questions
Top CAT Geometry Questions
- Let $ABCDEF$ be a regular hexagon and $P$ and $Q$ be the midpoints of $AB$ and $CD$, respectively. Then, the ratio of the areas of trapezium $PBCQ$ and hexagon $ABCDEF$ is:
- ABCD is a trapezium in which AB is parallel to DC, AD is perpendicular to AB, and $AB = 3DC$. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is
- The $(x, y)$ coordinates of vertices $P$, $Q$ and $R$ of a parallelogram $PQRS$ are $(-3, -2)$, $(1, -5)$ and $(9, 1)$, respectively. If the diagonal $SQ$ intersects the x-axis at $(a, 0)$, then the value of $a$ is:
- In a circle with center $C$ and radius $6\sqrt{2}$ cm, $PQ$ and $SR$ are two parallel chords separated by one of the diameters. If $\angle PQC = 45^\circ$, and the ratio of the perpendicular distance of $PQ$ and $SR$ from $C$ is $3:2$, then the area, in sq. cm, of the quadrilateral $PQRS$ is:
- In a $\triangle ABC$, points $D$ and $E$ are on the sides $BC$ and $AC$, respectively. $BE$ and $AD$ intersect at point $T$ such that $AD : AT = 4 : 3$, and $BE : BT = 5 : 4$. Point $F$ lies on $AC$ such that $DF$ is parallel to $BE$. Then, $BD : CD$ is:
View More Questions
Top CAT Questions
- The set of all real values of $x$ for which $(x^2 - |x+9| + x)>0$ is:
- CAT - 2025
- Number System
- For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is
- CAT - 2025
- Number System
- The rate of water flow through three pipes A, B and C are in the ratio 4 : 9 : 36. An empty tank can be filled up completely by pipe A in 15 hours. If all the three pipes are used simultaneously to fill up this empty tank, the time, in minutes, required to fill up the entire tank completely is nearest to
- CAT - 2025
- Time and Work
- The ratio of expenditures of Lakshmi and Meenakshi is $2 : 3$, and the ratio of income of Lakshmi to expenditure of Meenakshi is $6 : 7$. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is $4 : 9$, then the ratio of their incomes is:
- The average salary of 5 managers and 25 engineers in a company is 60000 rupees. If each of the managers received 20% salary increase while the salary of the engineers remained unchanged, the average salary of all 30 employees would have increased by 5%. The average salary, in rupees, of the engineers is
- CAT - 2025
- Averages
View More Questions