Question

14. The perimeter of an isoseles triangle is 42 em and its base is 15 times
each of the equal sides. Find (i) the length of each side of the triangle,
(11) the area of the triangle, and (ii) the height of the triangle. (Given,
√7=
264)​

Answer 1

QUESTION:

The perimeter of an isosceles triangle is 42 cm and its base is 1.5 times each of the equal sides. Find

(i) the length of each side of the triangle,

(ii) the area of the triangle, and

(iii) the height of the triangle.

(Given, √7=2.64)

ANSWER:

Given:

• Perimeter of isosceles triangle = 42cm
• Base = 1.5 times equal side
• √7 = 2.64

To Find:

• Length of each side of triangle
• Area of triangle
• Height of triangle

Solution:

$\text{1. Let the length of each equal side be x cm.}\\\\\text{So,}\\\\:\implies\sf Base=1.5\times x=1.5x\\\\\text{We are given that,}\\\\:\longrightarrow \sf Perimeter=42cm\\\\\text{So,}\\\\:\implies\sf42=x+x+1.5x\\\\:\implies\sf3.5x=42\\\\:\implies\sf x=\dfrac{42}{3.5}\\\\:\implies\sf x=12cm\\\\\text{So,}\\\\:\implies\sf Base=1.5\times x=1.5\times 12=18cm\\\\\text{\bf{So, the length of sides of triangle are 12cm, 12cm and 18cm respectively.}}$

$\text{2. We know that,}\\\\:\hookrightarrow\sf Area=\sqrt{s(s-a)(s-b)(s-c)}\\\\\text{Here, s is semi-perimeter and a,b,c are sides.}\\\\\text{So,}\\\\:\implies\sf s=\dfrac{42}{2}=21\\\\\text{Now,}\\\\:\implies\sf Area=\sqrt{s(s-a)(s-b)(s-c)}\\\\:\implies\sf Area=\sqrt{21(21-12)(21-12)(21-18)}\\\\:\implies\sf Area=\sqrt{21\times9\times9\times3}\\\\:\implies\sf Area=\sqrt{3\times7\times9\times9\times3}$

$:\implies\sf Area=\sqrt{3^2\times9^2\times7}\\\\:\implies\sf Area=3\times9\sqrt{7}\\\\:\implies\sf Area=27\times2.64\\\\:\implies\sf Area=71.28cm^2\\\\\text{\bf{Area of the triangle is 71.28cm ^2 }}$

$\text{3. We know that,}\\\\:\hookrightarrow\sf Area=\dfrac{1}{2}\times Base\times Height\\\\\text{Substituting the values of Area and Base,}\\\\:\implies\sf71.28=\dfrac{1}{2\!\!\!/}\times18\!\!\!\!\!/^{\:\:9}\times Height\\\\:\implies\sf Height=\dfrac{71.28}{9}\\\\\:\implies\sf Height=7.92cm\\\\\text{\bf{Height of the triangle is 7.92cm}}$

Formulae Used:

$:\hookrightarrow1) \sf Area\:of\:Triangle=\sqrt{s(s-a)(s-b)(s-c)}\\\\:\hookrightarrow2)\sf Area\:of\:Triangle=\dfrac{1}{2}\times Base\times Height$