Question:
Consider the triangle $ABC$ where $BC = 12$ cm, $DB = 9$ cm, $CD = 6$ cm, and $\angle BCD = \angle BAC$.
What is the ratio of the perimeter of $\triangle ADC$ to that of $\triangle BDC$?
Consider the triangle $ABC$ where $BC = 12$ cm, $DB = 9$ cm, $CD = 6$ cm, and $\angle BCD = \angle BAC$.
What is the ratio of the perimeter of $\triangle ADC$ to that of $\triangle BDC$?
Show Hint
Use similarity to relate corresponding sides, then sum for perimeters.
Updated On: Jul 31, 2025
- $\frac{7}{9}$
- $\frac{8}{9}$
- $\frac{6}{9}$
- $\frac{5}{9}$
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Given $\angle BCD = \angle BAC$, triangles $ADC$ and $CBD$ are similar by AA similarity (common angle at $C$).
From similarity:
\[
\frac{AD}{DB} = \frac{DC}{BC}
\]
$DB = 9$, $BC = 12$, $DC = 6$:
\[
\frac{AD}{9} = \frac{6}{12} \Rightarrow AD = 4.5
\]
Perimeter of $\triangle ADC$: $AD + DC + AC$. From similarity, $\frac{AC}{BC} = \frac{AD}{DB} = \frac12$, so $AC = 6$. Perimeter ADC = $4.5 + 6 + 6 = 16.5$.
Perimeter of $\triangle BDC$: $BD + DC + BC = 9 + 6 + 12 = 27$.
Ratio = $\frac{16.5}{27} = \frac{8}{9}$. \[ \boxed{\frac{8}{9}} \]
Perimeter of $\triangle BDC$: $BD + DC + BC = 9 + 6 + 12 = 27$.
Ratio = $\frac{16.5}{27} = \frac{8}{9}$. \[ \boxed{\frac{8}{9}} \]
Was this answer helpful?
0
0
Top Questions on Geometry
- A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
- If \( A(2, 3) \) and \( B(4, 5) \) are the end points of a straight line AB, then the slope of straight line AB is:
- Spheres of unit diameter are centered at \( (l, m, n) \), where \( l, m, \) and \( n \) take every possible integer value. The distance between two spheres is computed from the center of one sphere to the center of another sphere. For a given sphere, \( x \) is the distance to its nearest sphere and \( y \) is the distance to its next nearest sphere. The value of \( \frac{y}{x} \) is:
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.
View More Questions
Questions Asked in CAT exam
- ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of \(\triangle ADE\) is
- CAT - 2024
- Mensuration
- Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
- CAT - 2024
- Time, Speed and Distance
- The selling price of a product is fixed to ensure 40% profit. If the product had cost 40% less and had been sold for 5 rupees less, then the resulting profit would have been 50%. The original selling price, in rupees, of the product is
- CAT - 2024
- Profit and Loss
- Suppose $x_1, x_2, x_3, \dots, x_{100}$ are in arithmetic progression such that $x_5 = -4$ and $2x_6 + 2x_9 = x_{11} + x_{13}$. Then, $x_{100}$ equals ?
- CAT - 2024
- Arithmetic Progression
- A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of $120^\circ$ at the center. Then, the area, in square meters, of the smaller region is
- CAT - 2024
- Mensuration
View More Questions