Question

Area of an isosceles triangle is 48 sq. cm. If the altitudes corresponding to the base of the triangle is 8 cm, find the perimeter of the triangle. (Single choice.)


(a) 36 cm

(b) 32 cm

(c) 48 cm

(d) 96 cm​

Answer 1

Answer :

• The perimeter of the isosceles triangle is 32 cm.

Explanation :

Given :

• Area of the isosceles triangle, A = 48 cm²
• Height corresponding to the base of the isosceles triangle, h = 8 cm

To find :

• The perimeter of the isosceles triangle, p = ?

Knowledge required :

• Pythagoras theorem :

By definition, Pythagoras theorem says that the the sum of squares of the base and height of a right-angled triangle is equal to the square of hypotenuse of that triangle.

Mathematically,

Pythagoras theorem : h² = b² + p²

[Where : h, b and p are the hypotenuse, base and height of the triangle, respectively.]

• Area of a triangle :

By definition, the area of a triangle is the half of the product of the height and base of the triangle.

Mathematically,

Area of a triangle : ½ × b × h

[Where : b and h are the base and height of the triangle, respectively.]

• Perimeter of a triangle :

By definition, the perimeter of the triangle is the sum of all the three sides of the triangle.

Mathematically,

Perimeter of a triangle : a + b + c

[Where : a, b and c are the sides of the triangle]

Solution :

To find the base of the triangle :

⠀By using the formula for area of a triangle and substituting the values in it, we get :

⠀⠀=> ½ × b × h = A

⠀⠀=> ½ × b × 8 = 48

⠀⠀=> b × 4 = 48

⠀⠀=> b = 48/4

⠀⠀=> b = 12

⠀⠀⠀∴ b = 12 cm

⠀Hence the base of the triangle is 12 cm.

To find the hypotenuse of the triangle (s) formed by the altitude.

By using the pythagoras theorem and substituting the values in it, we get :

⠀⠀=> h² = b² + p²

⠀⠀=> h² = (12/2)² + 8²

⠀⠀=> h² = 6² + 8²

⠀⠀=> h² = 36 + 64

⠀⠀=> h² = 100

⠀⠀=> h = √100

⠀⠀=> h = 10

⠀⠀⠀∴ h = 10 cm

Hence the hypotenuse of the triangle is 10 cm.

To find the perimeter of the triangle :

⠀By using the formula for perimeter of a triangle and substituting the values in it, we get : (a = 12 cm, b = 10 cm]

⠀⠀=> P = a + b + c [b = c, since it's a isosceles triangle]

⠀⠀=> P = 12 + 10 + 10

⠀⠀=> P = 32

⠀⠀⠀∴ P = 32 cm

Therefore,

• Perimeter of the triangle, p = 32 cm

Answer 2

${\bold{\sf{\underline{Question}}}}$

✰ Area of an isosceles triangle is 48 sq. cm. If the altitudes corresponding to the base of the triangle is 8 cm, find the perimeter of the triangle. Options are given as : 36 cm , 32 cm , 48 cm and 96 cm.

${\bold{\sf{\underline{Given \: that}}}}$

✰ Area of isoscles triangle = 48 cm²

✰ The altitudes / height corresponding to the base of the triangle is 8 cm.

${\bold{\sf{\underline{To \: find}}}}$

✰ Perimeter of given triangle.

${\bold{\sf{\underline{Solution}}}}$

✰ Perimeter of given triangle = 32 cm

${\bold{\sf{\underline{Understanding \: the \: question}}}}$

❖ This question says that there is a triangle given and it's an isoscles triangle and it's area is 48 cm² Now it's altitudes corresponding to the base of the triangle is 8 cm. Altitude means height. Now this question ask us to find the perimeter of the triangle.

${\bold{\sf{\underline{Using \: concepts}}}}$

❖ Pythagoras theorem formula

❖ Area of triangle formula.

❖ Perimeter of triangle formula.

${\bold{\sf{\underline{Using \: formulas}}}}$

❖ Pythagoras theorem = (Height)² = (Base)² + (Perpendicular)²

❖ Area of triangle = ½ × Base × Height

❖ Perimeter of triangle = 1st side + 2nd side + 3rd side

${\bold{\sf{\underline{These \: also \: means}}}}$

❖ Area as A

❖ Base as B

❖ Height as H

❖ Perpendicular as p

❖ ½ means half

${\bold{\sf{\underline{Procedure \: of \: the \: question}}}}$

✰ To solve this problem firstly we have to find the base of ∆. Putting the values we get 12 cm as base of ∆. Afterthat we have to find the hypotenuse corresponding to the altitude / height. Substituting the value. we get 10 cm as the h. Now we have to find the perimeter of ∆ by using the formula to find the perimeter of ∆ substituting the values we get 32 cm as final result.

${\bold{\sf{\underline{Full \: solution}}}}$

Finding the base [ Putting the values ]

➨ A = ½ × B × H

➨ 48 = ½ × B × 8

➨ 48 = 1 × B × 4

➨ 48 = 4 × B

➨ 48 / 4 = B

➨ 12 = B

➨ B = 12 cm

• Henceforth, base is 12 cm

Finding the hypotenuse corresponding to the altitude / height [ Putting the values ]

➨ (Height)² = (Base)² + (Perpendicular)²

➨ (Height)² = (12/2)² + (8)²

➨ (Height)² = (6)² + (8)²

➨ (Height)² = 36 + 64

➨ (Height)² = 100

➨ Height = √100

➨ Height = 10 cm

• Henceforth, height is 10 cm.

Finding the perimeter of ∆

➨ P = 1st side + 2nd side + 3rd side

➨ P = 12 + 10 + 10

➨ P = 12 + 20

➨ P = 32 cm

• Henceforth, perimeter is 32 cm.

${\bold{\sf{\underline{Knowledge \: booster}}}}$

${\bold{\sf{Phythagoras \: theorm}}}$ - This theorm state that the sum of squares of the base and the height of a right angled triangle is equal to square of hypotenuse. Hypotenuse is also known as ${\bold{\pink{\sf{perpendicular}}}}$

${\bold{\sf{Triangle}}}$ - It's a 2D shape that have 3 angles, 3 sides 3 vertices, When 3 sides form and meet , they create three vertices, or corners and it's known to be a triangle. 2D means 2 dimensional. The symbol to show triangle is ∆

There are ${\bold{\pink{\sf{3}}}}$ type of triangle : Equaliteral triangle , Isoscles triangle and Scalene triangle.

${\bold{\pink{\sf{Equaliteral \: triangle}}}}$ - This ∆ have all equal sides that's why it's known as Equilateral triangle.

${\bold{\pink{\sf{Isosceles \: triangle}}}}$ - This ∆ have 2 equal sides and 1 unequal to 2 other sides

${\bold{\pink{\sf{Scalene \: triangle}}}}$ - This ∆ have all unequal sides.