Answer

Verified

480.9k+ 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

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Harsha Charita was written by A Kalidasa B Vishakhadatta class 7 social science CBSE

Which are the Top 10 Largest Countries of the World?

Banabhatta wrote Harshavardhanas biography What is class 6 social science CBSE

Difference Between Plant Cell and Animal Cell

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

How do you graph the function fx 4x class 9 maths CBSE

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

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE