Question

Find the base of an isosceles triangle whose area is 60m2 and length of equal sides is 13 cm

Answer 1

$\large{\underline{\red{\rm{AnswEr:Base=24m}}}}$

$\bold\green{\underline{Given:}}$

The area of isosceles Triangle=60m²

The length of each sides=13cm

$\bold\orange{\underline{Find:The\:base\:of \: Triangle}}$

$\huge\bold\red{\underline{Solution:}}$

$\bold\green{\underline{Construction:}}$

Draw a perpendicular AD which is on BC with height H that's why BD=CD in isoceles traiangle ABD

Now , using pathagoras theorem

AB²=AD²+BD²

Here AB=60 AD=h BD=X

13²=h²+X²-------1

Similarly, find the area of traiangle ABC

Area=1/2×base×height

60=1/2×2X×h

120=2Xh------(2)

Adding equation 1 and 2 here

13²+120=h²+X²+2Xh

here , using identity

(a+b)²=a²+b²+2ab

169+120=(h+X)²

289=(h+X)²

√289=h+X

h+X=17

Solving it now

h=17-x

Now put the value of X in EQ 2

120=2X(17-X)

120=34X-2X²

Dividing by 2 on both sides here

X²-17X+60=0

Splitting the middle term here

X²-12X-5X+60=0

X(X-12)-5(X-12)=0

(X-5)(X-12)=0

SO,X=5 , 12

Now, putting the value of X=12 in EQ 2

120=2Xh

120=24h

h=5m

Here h=AD=5m BC=BD+CD=2X=2×12=24m

Hence,

The perpendicular of traiangle=5m

The base of isoceles traiangle=24m

Answer 2

AnswEr:-

Base of isosceles triangle =10 cm or 24 cm

$\rule{200}{2}$

Given:-

• Area of isosceles ∆ = 60 cm²

• Length of each equal sides = 13 cm

To find:-

• Base of isosceles triangle = ?

Solution:-

As we know,

☛ $\boxed{\boxed{\sf\green{Area \: of \: isosceles \: \triangle = \dfrac{b}{4}\sqrt{4a^2 - b^2} }}}$

Here, a = each equal side & b = inequal side(base)

• Putting all values :-

⇒ 60 = (b/4) √[(4 × 13²) - b²]

⇒ 60 = (b/4) √[4 × 169 - b²]

⇒ 60 = (b/4) √[676 - b²]

⇒ 60 = (b/4) √[26² - b²]

⇒ 60 × 4/b = √[676 - b²]

⇒ 240/b = √[676 - b²]

⇒ (240/b)² = 676 - b²

⇒ 57600/b² = 676 - b²

⇒ 57600 = 676b² - b⁴

⇒ 57600 - 676b² + b⁴ = 0

⇒ b⁴ - 676b² + 57600 = 0

⇒ b⁴ - 100b² - 576b² + 57600 = 0

⇒ b²(b² - 100) - 576(b² - 100) = 0

⇒ (b² - 100)(b² - 576) = 0

⇒ b² = 100 or b² = 576

⇒ b = √100 or b =√576

⇒ b = 10 or, b = 24

Therefore,

Base of isosceles triangle is 10 cm or 24 cm.