Question

Identical triangles can be joined together to form larger triangles that are

similar to each of the smaller ones, as shown below.
a) A larger triangle with a height of 18 cm is made in this way from
smaller triangles with a height of 3 cm and an area of 5.1 cm².
What is the area of this larger triangle?
b) Another larger triangle is made from smaller, similar triangles. The
perimeter of this larger triangle is 32 times larger than the perimeter of
one of the smaller triangles.
How many smaller triangles are used to form this larger triangle?

Answer 1

The area of the triangle and the number of triangles that form the large triangle obtained using the scale factor for length and area are;

a) 183.6 cm²

b) 1,024 small triangles

What is a scale factor?; Scale factor is the ratio of the lengths of the corresponding sides of an object and another object.

The question is with regards to scale factors;

The scale factor of area = The square of the scale factor of length

a) The height of the larger triangle = 18 cm

Height of the smaller triangle = 3 cm

Therefore, the scale factor of length is; 18 : 3 = 6 : 1

The length of a side on the larger triangle is 6 times the length of a side on the small triangle, therefore;

The area of the larger triangle is the product of the square of the scale factor of length and the area of the small triangle

Area of the larger triangle is; (6/1)² × 5.1 = 183.6 cm²

b) The perimeter of the larger triangle = 32 × The perimeter of the smaller triangles

Therefore; The scale factor of length is 32 : 1 = 32/1

The scale factor of area is; (32/1)² = 1,024/1

The area of the larger triangle is 1,024 times the area of the smaller triangle, therefore, the number of small triangles that form the large triangle are 1,024 small triangles