Question

Each of the equal sides of an isosceles triangle measures 2 cm more than its height, and the base ot the triangle measures 12 cm. Find the area of the triangle.​

Answer 1

✬ Area = 48 cm² ✬

Step-by-step explanation:

Given:

• Measure of both equal sides of an isosceles triangle is 2 cm more than its height.
• Measure of base of the triangle is 12 cm.

To Find:

• What is the area of triangle ?

Solution: Let the height of the triangle be x cm. Therefore,

➟ First side = 2 cm more than x.

• (x + 2) cm

➟ Second side = 2 cm more than x

• (x + 2) cm

Now, in right angled ∆EXF applying Pythagoras theorem.

★ H² = Perpendicular² + Base² ★

• EX (hypotenuse) = x + 2

• FX (base) = 6

• EF (perpendicular) = x

$\implies{\rm }$ EX² = EF² + FX²

$\implies{\rm }$ (x + 2)² = x² + 6²

$\implies{\rm }$ x² + 2² + 4x = x² + 36

$\implies{\rm }$ 4 + 4x = 36

$\implies{\rm }$ x = 36 – 4/4

$\implies{\rm }$ x = 32/4 = 8

So, we got

➮ Height of isosceles triangle = x = 8 cm

➮ Equal sides = x + 2 = 8 + 2 = 10 cm

Let's find the area of ∆EGX

★ Area of ∆ = 1/2 × Base × Height ★

• Base = GX = 12 cm

➨ (1/2 × GX × EF) cm²

➨ (1/2 × 12 × 8)

➨ (6 × 8)

➨ 48 cm²

Hence, area of isosceles triangle is 48 cm².

Answer 2

Given:–

• Each of the equal sides of an isosceles triangle measures 2 cm more than its height
• Base of the triangle is 12 cm

To Find:–

• What is the area of the triangle ?

Solution:–

Let an isosceles triangle with height 'x' cm and equal sides of triangle be 'x + 2' cm

• BD = DC = 6 cm

✧ In triangle ABD,

Applying Pythagoras theorem

❮ Hypotenuse² = Base² + Perpendicular ²❯

$\sf{\longmapsto AB^2 = BD^2 + AD^2 }$

$\sf{\longmapsto (x+2)^2 = 6^2 + x^2 }$

$\sf{\longmapsto \cancel{x^2} + 4x + 4 = 36 + \cancel{x^2} }$

$\sf{\longmapsto 4x = 36 -4}$

$\sf{\longmapsto 4x = 32 }$

$\sf{\longmapsto x = \cancel\dfrac{32}{4} }$

${\large{\underbrace{\bf{\star \: x = 8 \: \: \star}}}}$

• Height = x = 8 cm

Now,

To find the area of triangle, we use the formula:-

$\large{\boxed{\bf{\star \: Area = \dfrac{1}{2} \times Base \times Height \: \star}}}$

${\underline{\bf{According \: To \: Question:-}}}$

$\sf{\longmapsto Area = \dfrac{1}{\cancel{2}} \times \cancel{12} \times 8 }$

$\sf{\longmapsto Area = 6 \times 8}$

${\large{\underbrace{\bf{\star \: Area = 48 \: cm^2 \: \star}}}}$

Hence, the area of triangle is 48 cm²

━━━━━━━━━━━━━━━━━━━━━━━━━━━