D is a point on side B C of A B C such that, A D C= B A C. Show t… - Vidyadip

Question

$D$ is a point on side $B C$ of $\triangle A B C$ such that, $\angle A D C=\angle B A C$. Show that $A C^2=B C \times D C$.

✓

Answer

To Prove: $A C^2=B C \times D C$
Consider $\triangle A D C$ and $\triangle B A C$.
$\angle A D C=\angle B A C$(Given)
Also,
$\angle A C D=\angle B C A \quad$ (Common angle)
Therefore,
$\triangle A D C \sim \triangle B A C$ (by AA similarity test)
Hence, the corresponding sides of similar triangles are proportional.
$\frac{A C}{B C}=\frac{D C}{A C}$
On cross-multiplying, we get
$A C^2=B C \times D C$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from P-2 Similarity→P-2 Similarity — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

O is the centre of a circle of a radius $8\ cm$ the tangent at a point A on the circle cuts a line through O at B such that AB = $15\ cm$. Find OB.

If the HCF of 657 and 963 is expressible in the form 657x + 963x - 15, find x.

A solid cone of base radius 10cm is cut into two parts through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.

Prove that $\frac{1+\sec A }{\sec A }=\frac{\sin ^2 A }{1-\cos A }$

Can two numbers have 16 as their H.C.F. and 380 as their L.C.M? Give reason.

Evaluate the following:
If A and B are acute angle such that $\tan\text{A}=\frac13,\tan\text{B}=\frac12$ and $\tan(\text{A}+\text{B})=\frac{\tan\text{A}+\tan\text{B}}{1-\tan\text{A}\tan\text{B}},$ show that $\text{A}+\text{B}=45^\circ.$

Solve the following simultaneous equation graphically.
x + y = 0; 2x – y = 9

Find the mean marks of the student from the following cumulative frequency distribution.

Marks	Below 10	Below 20	Below 30	Below 40	Below 50	Below 60	Below 70	Below 80	Below 90
No. of students	5	9	17	29	45	60	70	78	83

The radii of the circular ends of a frustum of height 6cm are 1cm and 6cm, respectively. Find the slant height of the frustum.

Very-Short and Short-Answer Questions.If $\text{x}=\text{a}\sin\theta$ and $\text{y}=\text{b}\cos\theta,$ write the value of $\Big(\text{b}^2\text{x}^2+\text{a}^2\text{y}^2\Big).$