Question

an isosceles triangle with base 10cm has an area of 60cm². find its perimeter. 34 cm 26 cm 31 cm 36 cm

Explanation:

Step1: Find the height of the triangle

The area formula for a triangle is \( A = \frac{1}{2} \times base \times height \). Given \( A = 60 \, cm^2 \) and \( base = 10 \, cm \), we solve for height (\( h \)):
\[
60=\frac{1}{2}\times10\times h
\]
\[
60 = 5h
\]
\[
h=\frac{60}{5}=12 \, cm
\]

Step2: Find the length of the equal sides

In an isosceles triangle, the height splits the base into two equal parts. So each part is \( \frac{10}{2}=5 \, cm \). Using the Pythagorean theorem (\( a^2 + b^2 = c^2 \)) where \( a = 5 \), \( b = 12 \), and \( c \) is the length of the equal side:
\[
c=\sqrt{5^2 + 12^2}=\sqrt{25 + 144}=\sqrt{169}=13 \, cm
\]

Step3: Calculate the perimeter

The perimeter (\( P \)) of a triangle is the sum of all sides. For this isosceles triangle, two sides are \( 13 \, cm \) and the base is \( 10 \, cm \):
\[
P = 13 + 13 + 10 = 36 \, cm
\]
Wait, no, wait. Wait, 13+13+10 is 36? Wait, no, wait, 132 is 26, plus 10 is 36? Wait, but let's check again. Wait, height is 12, half base is 5, so the equal side is sqrt(5² +12²)=13. Then perimeter is 13+13+10=36? But wait, the options have 34,26,31,36. Wait, maybe I made a mistake? Wait, no, let's recalculate. Wait, area is 60, base 10, so height is (260)/10=12. Correct. Then half base is 5, so the equal side is sqrt(5² +12²)=13. So perimeter is 13+13+10=36. So the correct answer is 36 cm.

Answer:

36 cm (corresponding to the option with 36 cm)