MCQ
$D$ is a point on side $\text{BC}$ of a $\triangle ABC$ such that $\angle ADC=\angle BAC$, then $CA^2=$
- A$B C \times A D$
- B$B C^2$
- C$A B^2$
- ✓$B C \times C D$
✓
Answer
Correct option: D.
$B C \times C D$
In $\triangle A B C$ and $\triangle D A C$,
$\angle B A C=\angle A D C ($ Given$)$
$\angle B C A=\angle A C D($Common$)$
$\therefore \triangle A B C \sim \triangle D A C$
$($By $AA$ similarity criterion$)$
$\Rightarrow \frac{B C}{C A}=\frac{C A}{C D}$
$\Rightarrow C A^2=B C \times C D$
$\angle B A C=\angle A D C ($ Given$)$
$\angle B C A=\angle A C D($Common$)$
$\therefore \triangle A B C \sim \triangle D A C$
$($By $AA$ similarity criterion$)$
$\Rightarrow \frac{B C}{C A}=\frac{C A}{C D}$
$\Rightarrow C A^2=B C \times C D$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Equation of mean $X =a+\frac{\Sigma f_i u_i}{\Sigma f_i}$ where $u_i=$ _______
→If $9$ times the $9^{\text {th }}$ term in an arithmetic progression is equal to $15$ times of its $15^{\text {th }}$ term, then what is the $24^{\text {th }}$ term ?
→The $13^{th}$ term of an AP is 4 times its $3^{rd}$ term. If its $5^{th}$ term is $16$ then the sum of its first ten terms is:
→If three coins are tossed simultaneously, then the probability of getting at least two heads, is:
If $8\tan \text{x} = 15,$ then $\sin \text{x} - \cos \text{x}$ is equal to:
→If the polynomial $f(x)=a x^3+b x-c$ is divisible by the polynomial $g(x)=x^2+b x+c$, then $ab =$
→The graph of pair of Linear equations in two variables $2 x+3 y-9=0$ and $4 x+6 y-18=0$ is
→The volume of the greatest sphere that can be cut off from a cylindrical log of wood of base radius $1\ cm$ and height $5\ cm$ is:
→If two tangents inclined at a angle of $60^\circ $ are drawn to a circle of radius $3\ cm$, then length of each tangent is equal to:
→While computing the mean of the grouped data, we assume that the frequencies are: