Question

A triangle having semiperimeter 21 and one of the side is 13 and the difference between other two sides is 1. Area of the triangle (in sq. units) is​

Answer 1

Question

A triangle having semiperimeter 21 and one of the side is 13 and the difference between other two sides is 1. find the area of the triangle (in sq. units).

Solution

• Let the three sides of the triangle be a, b and c

• Semi-perimetre of the triangle is 21 units

According to the Question

⇒ a = 13 units

⇒ b-c = 1 units

∴ b = (c+1) units

As Per The Question :-

⇒$\large{\sf{Semi-perimetre=\frac{a+b+c}{2}}}$

⇒$\large{\sf{21=\frac{13+c+1+c}{2}}}$

⇒$\large{\sf{21×2=14+2c}}$

⇒$\large{\sf{42-14=2c}}$

⇒$\large{\sf{2c=28}}$

⇒$\large{\sf{c=\frac{28}{2}}}$

∴ c = 14 units

∴ b = (c+1) = (14+1) = 15 units

Answer

By Heron's Formula:-

⇒$\large{\sf{Area\:∆=\sqrt{s(s-a)(s-b)(s-c)}}}$

where,

• s = Semi-perimetre

• a = first side

• b = second side

• c = third side

★ Hence, putting the values we get →

⇒$\large{\sf{Area=\sqrt{21(21-13)(21-14)(21-15)}}}$

⇒$\large{\sf{Area=\sqrt{21×8×7×6}}}$

⇒$\large{\sf{Area=\sqrt{7056}}}$

⇒$\large{\sf{Area\:∆=84}}$

Therefore, the required area of the triangle is 84 units.

Answer 2

Let the side of triangle are a, b, c

acc. to question

• side a is 12
• Semi perimeter (S)is 21

$\: \: \: \: \: \: \large{ \sf \: S = \frac{a + b + c}{2} } \\ \bf \: \: \: \: \: \: 21 = \frac{13 + b + c}{2} \\ \bf\: \: \: \: \: 42 = 13 + b + c \\ \bf \: \: \: \: 42 - 13 = b + c \\ \bf \: \: \: \: \: \: \: \: \: \: \: \: \: \: 29 = b + c \: - - (1)$

The difference between the other two side is 1

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf1 = b - c \: \: \: - - (2)$

Now, equate eq. (1) and (2) , by adding

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 29 = b + \cancel{c} - - (1) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ + \: 1 = b - \cancel{c} }- - (2) \\30 = 2b \\\: \: \: \: b = \frac{30}{2} \\ \:\: \: \: \: \boxed{ \bf{b = 15}}$

By putting value of b in eq. (2)

$\: \: \: \: \: \: \: \: \: \bf \: b - c = 1 \\ \: \: \: \: \: \: \: \: \: 15 - c = 1 \\ \: \: \: \: \: \: \: \: \:15 - 1 = c \\ \: \: \: \: \: \: \: \: \: \boxed{\bf \: c = 14}$

$\boxed{\bf{A \:∆= \sqrt{S \: (S - a)(S - b)(S - c)} } }\\ \rightarrow \sqrt{21(21 - 13)(21 - 14)(21 - 15)} \\ \rightarrow \sqrt{21 \times 8 \times 7 \times 6} \\ \rightarrow \sqrt{7 \times 3 \times 2 \times 2 \times 2 \times 7 \times 3 \times 2} \\ \implies7 \times 3 \times 2 \times 2 \\ \implies21 \times 4 \\ \red{\star} \sf{A\: ∆} : \implies \: \boxed{ \sf84 \: sq. \: }$

∴ The required area of triangle is 84 units.

#Gunjan