Question

calculate the area of the triangle whose sides are 18 cm 24 cm and 30 cm in length also find the length of the altitude corresponding to the smallest side

Answer 1

Let's calculate the area first by Heron's Formula,


So first, we need to find s

so s =
$\frac{a + b + c}{2} \\ \\ = > \frac{18 + 24 + 30}{2} \\ \\ = > \frac{72}{2} = 36$


Now Heron's Formula =
$\sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{36(36 - 18)(36 - 24)(36 - 30)} \\ \\ = > \sqrt{36 \times 18 \times 12 \times 6} \\ \\ = > \sqrt{46656 } \\ \\ = 216$


So the area of triangle = 216 cm²


Now, area of triangle is also equal to

1/2 × b × h

Where b is the base and h is the height


Here, we have to find the height corresponding to smallest side.

Side = 18 cm

So taking base as 18 we get

1/2 × 18 × h = area

but area = 216

=> 1/2 × 18 × h = 216

=> 9h = 216

=> h = 216/9

=> h = 24 cm


So the height = 24 cm



Hope it helps dear friend ☺️✌️


Side note :- This means that the triangle is a right angled triangle, since the height = one of its side :)

Answer 2

$<b>Heya mate. (^_-). Solution below.$
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$<u>Your Question -</u>$

Calculate the area area of a triangle whose sides are 18 cm, 24 cm, and 30 cm in length. Also find the length of altitude corresponding to the smallest side.

$<u>Solution -</u>$

Here,
First side of the ∆, a = 18 cm
Second side, b = 24 cm
Third side, c = 30 cm

So, the perimeter of the triangle
= sum of all sides
= a + b + c
= 18 + 24 + 30
= 72 cm

Then, the semi perimeter, s
= Perimeter/2
= 72/2
= 36


So, by Heron's Formula,

Area of the ∆

= √{s(s-a)(s-b)(s-c)}

= √{36(36-18)(36-24)(36-30)}

= √{36 × 18 × 12 × 6}

= √46656

= √216 × 216

= 216 cm²

Hence, the area of the ∆ = 216 cm².


Now, the second part of the question.

The smallest side of the triangle = a = 18 cm

We also know that the area of a triangle = half the product of its base and its altitude.

So, the area of the triangle with base 18 cm

= ½ × b × h
= ½ × 18 × h
= 9×h
= 9h

But, the area of the triangle = 216 cm² (as proved above)

So,
9h = 216
=> h = 216/9
=> h = 24

So, the height = h = 24 cm.

$<marquee>The area of the triangle is 216 cm²</marquee>$

$<marquee>The height corresponding to the smallest side is 24 cm</marquee>$
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Thank you. ;-)