MCQ
$D$ is a point on side $B C$ of a $\triangle A B C$ such that $\angle A D C=\angle B A C$, then $C A^2=$
- 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
In a data, if $~l=60, h=15, f_1=16, f_0=6, f_2=6$, then the mode is :
→If origin is the mid-point of the line segment joining the points $P(a, b)$ and $Q(3,3)$, then the value of $(a+b)$ is
→A point on the $x$-axis divides the line segment joining the points $A(2,-3)$ and $B(5,6)$ in the ratio 1:2. The point is
→A rectangular sheet of paper 40cm × 22cm, is rolled to form a allow cylinder of height 40cm. The radius of e cylinder {in cth) is:
If $\triangle A B C \sim \triangle D E F$ such that $A B=9.1 cm$ and $D E=6.5 cm$. If the perimeter of $\triangle D E F$ is 25 cm , then the perimeter of $\triangle A B C$ is
→The ratio of the length of a pole and its shadow is $1: \sqrt{3}.$ The angle of elevation of the sun is :
→A solid sphere of radius $r$ is melted and cast into the shape of a solid cone of height $r$, the radius of the base of the cone is:
→The median of first 8 prime numbers is:
The roots of $a x^2+b x+c=0, a \neq 0$ are real and unequal, if $\left(b^2-4 a c\right)$ is :
→The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is :
→