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CAT 2023 Slot 1 QA Question 17

Multiple choice (+3 / −1) · Geometry · Quadrilaterals · Try it, then check the answer and solution below.

CAT 2023 Slot 1QAGeometry • QuadrilateralsHard
A quadrilateral ABCDA B C D is inscribed in a circle such that AB:CD=2:1A B: C D=2: 1 and BC:AD=5:4B C: A D=5: 4. If ACA C and BDB D intersect at the point E,thenE , thenA E::C E$ equals
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Answer and solution

Answer: B) 8:58:5

📌 Core Concept
The problem involves a cyclic quadrilateral where the diagonals intersect. The key theorem used is the Ratio Lemma for cyclic quadrilaterals, which states that the ratio of the segments of one diagonal is equal to the ratio of the products of the adjacent sides.
🔢 Step-by-Step Solution
1
Given Ratios:
AB:CD=2:1AB : CD = 2 : 1
BC:AD=5:4BC : AD = 5 : 4
2
Let:
AB=2x,CD=xAB = 2x , CD = x
BC=5y,AD=4yBC = 5y , AD = 4y
3
Apply the Ratio Lemma:
AEEC=AB×ADBC×CD\frac{AE}{EC} = \frac{AB \times AD}{BC \times CD}
4
Substitute the Values:
AEEC=2x×4y5y×x=8xy5xy=85\frac{AE}{EC} = \frac{2x \times 4y}{5y \times x} = \frac{8xy}{5xy} = \frac{8}{5}
5
Simplify:
AE:EC=8:5AE : EC = 8 : 5
⚡ 30-Second Shortcut
For cyclic quadrilaterals, use the Ratio Lemma: AEEC=AB×ADBC×CD\frac{AE}{EC} = \frac{AB \times AD}{BC \times CD}. Plug in the given ratios directly to find the segment ratio.
🎯 Final Answer
Correct Answer: Option B

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