Question

the altitude of a right triangle is 7 cm less than its bace . if the hypotinues is 13 cm , find the other two sides

Answer 1

» To Find :

The Base and Height of the Right-angled Triangle.

» Given :

• Hypotenuse = 13 cm

» We Know :

Pythagoras theorem :

$\sf{\underline{\boxed{h^{2} = p^{2} + b^{2}}}}$

Where ,

• h = hypotenuse of the triangle
• p = height of the triangle
• b = base of the triangle

» Concept :

Let the base be x.

According to the question , the height is 7 cm less than the base of the triangle ,so the Equation formed is (x - 7).

Now by using the Pythagoras theorem ,we can find the value of x, by which we can find the height and the base of the triangle.

» Solution :

By the above information , we got the values as ,

• Hypotenuse = 13 cm

• Base = x cm

• Height = (x - 7) cm

Pythagoras theorem :

$\sf{\underline{\boxed{h^{2} = p^{2} + b^{2}}}}$

By substituting the values in it ,we get :

$\sf{\Rightarrow 13^{2} = (x - 7)^{2} + x^{2}}$

By Using the indentity ,

$\sf{(a - b)^{2} = a^{2} - 2ab + b^{2}}$

we get :

$\sf{\Rightarrow 169 = x^{2} - 2 \times 7x + 7^{2} + x^{2}}$

$\sf{\Rightarrow 169 = x^{2} - 14x + 7^{2} + x^{2}}$

$\sf{\Rightarrow 169 = 2x^{2} - 14x + 49}$

$\sf{\Rightarrow 0 = 2x^{2} - 14x + 49 - 169}$

$\sf{\Rightarrow 0 = 2x^{2} - 14x - 120}$

By Using the middle-splitting theorem , we get :

$\sf{\Rightarrow 0 = 2x^{2} - (24 - 10)x - 120}$

$\sf{\Rightarrow 0 = 2x^{2} - 24x + 10x - 120}$

$\sf{\Rightarrow 0 = 2x(x - 12) + 10(x - 12)}$

Taking the common i.e. (x - 12) ,we get :

$\sf{\Rightarrow 0 = (x - 12)(2x + 10)}$

Hence ,the value of x is :

• $x - 12 = 0$

$\Rightarrow x = 12$

• $2x + 10 = 0$

$\Rightarrow 2x = - 10$

$\Rightarrow x = \dfrac{- 10}{2}$

$\Rightarrow x = \cancel{\dfrac{- 10}{2}}$

$\Rightarrow x = - 5$

Hence , the value of x is 12 and - 5 ,but as the side can't be less than 1 ,so we will neglect the - 5 part.

So ,the value of x is 12.

Hence ,the base of the triangle is 12 cm.

According to the question ,it is said that the height of the triangle is 7 cm less than the base if the triangle . i.e,

Height = Base - 7

Height = 12 - 7

Height = 5 cm

Thus , the height is 5 cm

So ,we get the Base as 12 cm and Height as 5 cm.

Additional information :

• Area of a triangle = ½ × base × height

• Diagonal of a Cube = √3a

• Surface area of a Cylinder = 2πr(h + r)

• Surface area of a Cuboid = 2(lb + lh + bh)

Answer 2

Given,

• Altitude of right triangle is 7 cm less than its base.
• Hypotenuse is 13 cm.

To find,

• The other two sides.

Solution,

Let x be the base of the triangle

Then altitude will be (x-7)

We know that,

$\sf{Base^2+Altitude^2=Hypotenuse^2}$

So, by pythagoras theorem,

${x}^{2} + ( {x - 7)}^{2} = {13}^{2} \\ \\ ⟹2 {x}^{2} - 14x + 49 = 169 \\ \\ ⟹2 {x}^{2} - 14x + 49 - 169 = 0 \\ \\ ⟹2 {x}^{2} - 14x - 120 = 0 \\ \\ ⟹2( {x}^{2} - 7x - 60) = 0 \\ \\ ⟹ {x}^{2} - 7x - 60 = \frac{0}{2} \\ \\⟹ {x }^{2} - 7x - 60 = 0 \\ \\ ⟹ {x}^{2} - 12x + 5x - 60 = 0 \\ \\ ⟹x(x - 12) + 5(x - 12) = 0 \\ \\ ⟹(x - 12)(x + 5) = 0$

So, x = 12 or x = -5

Since,the side of a triangle cannot be negative,so the base of the triangle is 12 cm.

And the altitude will be (12-7) = 5 cm