Answer
Verified
423.6k+ 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
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE