Question

The base of a right angled triangle is 8 m and its hypotenuse is 10 m. Then its area is
A) \[48{m^2}\]
B) \[40{m^2}\]
C) \[30{m^2}\]
D) \[24{m^2}\]

Answer 1

Hint:
Here, we have to find the area of the right triangle. First, we will use the Pythagoras theorem to find the height of a right-angled triangle. Then we will use the area of the right angled triangle formula to find the area. A right angled triangle is a triangle in which one of the angles is equal to \[90^\circ \].

Formula Used:
Area of the Right angled triangle is given by the formula \[A = \dfrac{1}{2} \times b \times h\], \[b\] is the base and \[h\] is the height of the triangle.

Complete step by step solution:
It is given that the base of a right angled triangle is 8 m and its hypotenuse is 10 m.

Let \[x\] be the height of the right angle triangle
Now, we will use the Pythagoras theorem to find the height of the triangle. Therefore, we get
\[A{B^2} = A{C^2} + B{C^2}\]
Rewriting the equation, we get
\[ \Rightarrow A{C^2} = A{B^2} - B{C^2}\]
Substituting \[AB = 10\] and \[BC = 8\] in the above equation, we get
\[ \Rightarrow {x^2} = {10^2} - {8^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {x^2} = 100 - 64\]
Subtracting the terms, we get
\[ \Rightarrow {x^2} = 36\]
Taking square root on both the sides, we get
\[ \Rightarrow x = \sqrt {36} \]
\[ \Rightarrow x = \pm 6\]
The height of the triangle cannot be negative. So,
\[ \Rightarrow x = 6m\]
Thus, the height of a triangle is 6 m.
Now, we will find the area of the given triangle.
Substituting \[b = 8\] and \[h = 6\]in the formula \[A = \dfrac{1}{2} \times b \times h\], we get
\[A = \dfrac{1}{2} \times 8 \times 6\]
Dividing the terms, we get
\[ \Rightarrow A = 4 \times 6\]
Multiplying the terms, we get
\[ \Rightarrow A = 24{{\rm{m}}^2}\]

Therefore, the area of the triangle is \[24{{\rm{m}}^2}\] and thus Option(D) is correct.

Note:
We know that the Pythagoras theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides. Pythagoras theorem can be applied only if the given triangle is a right angle triangle. Using this theorem, we will be able to find any side of the right-angled triangle. In a right-angled triangle, the hypotenuse is the longest side. The area of a triangle is nothing but the space covered by the triangle.