Answer

Verified

449.4k+ views

Hint: Draw the rough figure. Consider \[\Delta ABC\] and\[\Delta ADC\]. Prove they are similar by AA criteria. Using AA similarity, postulate if 2 triangles are similar then corresponding sides are proportional. Hence equate sides.

Complete step-by-step answer:

Let us draw a rough figure of triangle ABC such that D is a point of the side BC (i.e. base) of the triangle. Then join AD. It is given that \[\angle ADC\] is equal to \[\angle BAC\] which is given in the figure, i.e. \[\angle ADC=\angle BAC\].

We need to prove that \[C{{A}^{2}}=CB.CD\].

This can be rearranged and written as,

\[\begin{align}

& CA.CA=CB.CD \\

& \Rightarrow \dfrac{CA}{CD}=\dfrac{CB}{CA} \\

\end{align}\]

Now let us consider \[\Delta ABC\]and\[\Delta ADC\].

\[\angle ACB=\angle ACD\], they are common angles of both the triangles and it is given that\[\angle ADC=\angle BAC\].

Thus by AA similarity criterion we can say that the two triangles are similar.

Here in the AA criterion, two triangles are similar if all the corresponding angles are congruent and their corresponding sides will be proportional, i.e. by AA similarity postulate, if two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.

\[\therefore \Delta BAC\tilde{\ }\Delta ADC\].

We said that if two triangles are equal then their corresponding sides are proportional, which means we get,

\[\dfrac{BA}{AD}=\dfrac{AC}{CD}=\dfrac{BC}{AC}\]

Hence if we are taking, \[\dfrac{AC}{CD}=\dfrac{BC}{AC}\], i.e. \[\dfrac{CA}{CD}=\dfrac{CB}{CA}\].

We get by cross multiplying,

\[\begin{align}

& CA.CA=CB.CD \\

& \Rightarrow C{{A}^{2}}=CB.CD \\

\end{align}\]

Hence we proved what was asked in the question.

Note: The mathematical definition for similar triangles states that the triangles having proportional sides and all corresponding angles are the same.

If we consider there are two triangles, where each side of the larger triangle is twice as large as the smaller triangle, then by AA criteria, the triangles are similar if two corresponding angles are similar to each other.

Complete step-by-step answer:

Let us draw a rough figure of triangle ABC such that D is a point of the side BC (i.e. base) of the triangle. Then join AD. It is given that \[\angle ADC\] is equal to \[\angle BAC\] which is given in the figure, i.e. \[\angle ADC=\angle BAC\].

We need to prove that \[C{{A}^{2}}=CB.CD\].

This can be rearranged and written as,

\[\begin{align}

& CA.CA=CB.CD \\

& \Rightarrow \dfrac{CA}{CD}=\dfrac{CB}{CA} \\

\end{align}\]

Now let us consider \[\Delta ABC\]and\[\Delta ADC\].

\[\angle ACB=\angle ACD\], they are common angles of both the triangles and it is given that\[\angle ADC=\angle BAC\].

Thus by AA similarity criterion we can say that the two triangles are similar.

Here in the AA criterion, two triangles are similar if all the corresponding angles are congruent and their corresponding sides will be proportional, i.e. by AA similarity postulate, if two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.

\[\therefore \Delta BAC\tilde{\ }\Delta ADC\].

We said that if two triangles are equal then their corresponding sides are proportional, which means we get,

\[\dfrac{BA}{AD}=\dfrac{AC}{CD}=\dfrac{BC}{AC}\]

Hence if we are taking, \[\dfrac{AC}{CD}=\dfrac{BC}{AC}\], i.e. \[\dfrac{CA}{CD}=\dfrac{CB}{CA}\].

We get by cross multiplying,

\[\begin{align}

& CA.CA=CB.CD \\

& \Rightarrow C{{A}^{2}}=CB.CD \\

\end{align}\]

Hence we proved what was asked in the question.

Note: The mathematical definition for similar triangles states that the triangles having proportional sides and all corresponding angles are the same.

If we consider there are two triangles, where each side of the larger triangle is twice as large as the smaller triangle, then by AA criteria, the triangles are similar if two corresponding angles are similar to each other.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE