Question : In a square ABCD, diagonals AC and BD intersect at O. The angle bisector of $\angle CAB$ meets BD and BC at F and G, respectively. OF : CG is equal to:
Option 1: $1 : 2$
Option 2: $1 : 3$
Option 3: $1: \sqrt{2}$
Option 4: $1: \sqrt{3}$
Latest: SSC CGL preparation tips to crack the exam
Don't Miss: SSC CGL complete guide
New: Unlock 10% OFF on PTE Academic. Use Code: 'C360SPL10'
Correct Answer: $1: \sqrt{2}$
Solution : Given: ABCD is a square. Diagonal intersects at O. The angle bisector of $\angle$CAB meets BD and BC at F and G. Let's denote the side AB as \(x\). So, the diagonal AC = \(\sqrt{2}x\). $⇒OA = OB = OC = OD = \frac{AC}{2} = \frac{x}{\sqrt{2}}$ ___(i) Now, in $\triangle$AOB and from the angle bisector theorem, We have \(\frac{AB}{AO} = \frac{BF}{OF}\) \(⇒\frac{x}{\frac{x}{\sqrt{2}}} = \frac{BF}{OF}\) Let's denote BF as \(\sqrt{2}y\) and OF as \(y\). From equation (i), We know that \(BF + OF = OB = \frac{x}{\sqrt{2}}\) \(⇒\sqrt{2}y + y = \frac{x}{\sqrt{2}}\) \(⇒y = \frac{x}{\sqrt{2} + 2}\) \(⇒OF = \frac{x}{\sqrt{2} + 2}\) ___ (iii) In $\triangle$ABC and from the angle bisector theorem, We have \(\frac{AB}{AC} = \frac{BG}{GC}\) \(⇒\frac{x}{\sqrt{2}x} = \frac{BG}{GC}\) Let's denote BG as \(z\) and GC as \(\sqrt{2}z\).___ (iv) Since \(BG + GC = BC = x\) We have \(z + \sqrt{2}z = x\) \(⇒z = \frac{x}{\sqrt{2} + 1}\) \(⇒CG = \frac{2x}{\sqrt{2} + 1}\)___ (v) Now from equations (iii) and (v), We get \(OF : CG = \frac{x}{\sqrt{2} + 2} : \frac{2x}{\sqrt{2} + 1}\) \(OF : CG = 1 : 2\). Hence, the correct answer is $1:2$.
Candidates can download this ebook to know all about SSC CGL.
Answer Key | Eligibility | Application | Selection Process | Preparation Tips | Result | Admit Card
Question : Inside a square ABCD, $\triangle$BEC is an equilateral triangle. If CE and BD intersect at O, then $\angle$BOC is equal to:
Question : $ABCD$ is a cyclic quadrilateral of which $AB$ is the diameter. Diagonals $AC$ and $BD$ intersect at $E$. If $\angle DBC=35^\circ$, then $\angle AED$ measures _________
Question : In $\triangle \mathrm{ABC}, \angle \mathrm{A}=50^{\circ}$, BE and CF are perpendiculars on AC and AB at E and F, respectively. BE and CF intersect at H. The bisectors of $\angle \mathrm{HBC}$ and $\angle H C B$ intersect at P. $\angle B P C$ is equal to:
Question : ABCD is a quadrilateral in which BD and AC are diagonals. Then, which of the following is true:
Question : A cyclic quadrilateral ABCD is such that AB = BC, AD = DC and AC and BD intersect at O. If $\angle C A D=46^{\circ}$, then the measure of
Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile