In the given figure, ABC is an isosceles right triangle, right-angled at C. Prove that $A{{B}^{2}}=2A{{C}^{2}}$.

Answer
360.3k+ views
Hint: Two sides of an isosceles triangle is equal. Pythagoras theorem can be given as ${{\left( \text{hypotaneous} \right)}^{\text{2}}}\text{=}{{\left( \text{base} \right)}^{\text{2}}}\text{+}{{\left( \text{perpendicular} \right)}^{\text{2}}}$. Using these two things the desired result can be obtained.
Complete step-by-step answer:
We have an isosceles triangle ABC with$\angle C={{90}^{o}}$ and hence sides AC and BC are equal to each other, as in isosceles triangle two sides are equal. Hence, diagram of $\Delta ABC$can be represented as
Where AC = BC and $\angle C={{90}^{o}}$
Now, we need to prove the relation given as
$A{{B}^{2}}=2A{{C}^{2}}$ ………………….. (i)
So, let us calculate the value of $A{{B}^{2}}$in terms of $A{{C}^{2}}$to prove the above equation (i).
As we know any right angle triangle will follow the Pythagoras property which can be given as
${{\left( \text{hypotaneous} \right)}^{\text{2}}}\text{=}{{\left( \text{base} \right)}^{\text{2}}}\text{+}{{\left( \text{perpendicular} \right)}^{\text{2}}}$………… (iii)
Hence, we can write the equation (iii) in terms of sides of $\Delta ABC$ as
${{\left( AB \right)}^{2}}={{\left( BC \right)}^{2}}+{{\left( AC \right)}^{2}}$…………… (iv)
Now, as the given triangle is isosceles, therefore AC=BC from the figure; and hence, we can replace BC by AC in the equation (iv), so, we get
$\begin{align}
& {{\left( AB \right)}^{2}}={{\left( AC \right)}^{2}}+{{\left( AC \right)}^{2}} \\
& \Rightarrow {{\left( AB \right)}^{2}}=2{{\left( AC \right)}^{2}} \\
\end{align}$
Hence, equation (i) or the given relation is proved.
Note:
One can go wrong with the Pythagoras theorem. One may put the value of base or perpendicular in place of hypotenuse or vice-versa may also happen. Hence, be clear with the terms of the Pythagoras theorem.
One can get angles A and B as ${{45}^{o}}$. As both are equal and summation of all the angles of the triangle is ${{180}^{0}}$. Now, take $\sin {{45}^{o}}$in the triangle with respect to angle B.
Hence $\sin B=\sin {{45}^{o}}=\dfrac{AC}{AB}$
Now, put $\sin {{45}^{o}}=\dfrac{1}{\sqrt{2}}$ and square both sides.
We will get the same result as given in the question.
Complete step-by-step answer:
We have an isosceles triangle ABC with$\angle C={{90}^{o}}$ and hence sides AC and BC are equal to each other, as in isosceles triangle two sides are equal. Hence, diagram of $\Delta ABC$can be represented as

Where AC = BC and $\angle C={{90}^{o}}$
Now, we need to prove the relation given as
$A{{B}^{2}}=2A{{C}^{2}}$ ………………….. (i)
So, let us calculate the value of $A{{B}^{2}}$in terms of $A{{C}^{2}}$to prove the above equation (i).
As we know any right angle triangle will follow the Pythagoras property which can be given as
${{\left( \text{hypotaneous} \right)}^{\text{2}}}\text{=}{{\left( \text{base} \right)}^{\text{2}}}\text{+}{{\left( \text{perpendicular} \right)}^{\text{2}}}$………… (iii)
Hence, we can write the equation (iii) in terms of sides of $\Delta ABC$ as
${{\left( AB \right)}^{2}}={{\left( BC \right)}^{2}}+{{\left( AC \right)}^{2}}$…………… (iv)
Now, as the given triangle is isosceles, therefore AC=BC from the figure; and hence, we can replace BC by AC in the equation (iv), so, we get
$\begin{align}
& {{\left( AB \right)}^{2}}={{\left( AC \right)}^{2}}+{{\left( AC \right)}^{2}} \\
& \Rightarrow {{\left( AB \right)}^{2}}=2{{\left( AC \right)}^{2}} \\
\end{align}$
Hence, equation (i) or the given relation is proved.
Note:
One can go wrong with the Pythagoras theorem. One may put the value of base or perpendicular in place of hypotenuse or vice-versa may also happen. Hence, be clear with the terms of the Pythagoras theorem.
One can get angles A and B as ${{45}^{o}}$. As both are equal and summation of all the angles of the triangle is ${{180}^{0}}$. Now, take $\sin {{45}^{o}}$in the triangle with respect to angle B.
Hence $\sin B=\sin {{45}^{o}}=\dfrac{AC}{AB}$
Now, put $\sin {{45}^{o}}=\dfrac{1}{\sqrt{2}}$ and square both sides.
We will get the same result as given in the question.
Last updated date: 21st Sep 2023
•
Total views: 360.3k
•
Views today: 4.60k
Recently Updated Pages
What do you mean by public facilities

Slogan on Noise Pollution

Paragraph on Friendship

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

What is the Full Form of ILO, UNICEF and UNESCO

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

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

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE
