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

Feb 4, 2024

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

16 \mathrm{~cm}

is made in this way from smaller triangles with a height of

2 \mathrm{~cm}

and an area of

2.3 \mathrm{~cm}^{2}

. 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?

Solution by Steps

step 1

Determine the scale factor between the heights of the larger and smaller triangles

step 2

The scale factor is the ratio of the larger triangle's height to the smaller triangle's height:

\frac{16 \text{ cm}}{2 \text{ cm}} = 8

step 3

The area of similar triangles scales with the square of the scale factor

step 4

Calculate the area of the larger triangle using the scale factor squared:

\text{Area}_{\text{larger}} = \text{Area}_{\text{smaller}} \times (\text{scale factor})^2

step 5

Substitute the given values to find the area of the larger triangle:

\text{Area}_{\text{larger}} = 2.3 \text{ cm}^2 \times 8^2

step 6

Compute the area:

\text{Area}_{\text{larger}} = 2.3 \text{ cm}^2 \times 64

step 7

Simplify to get the final area:

\text{Area}_{\text{larger}} = 147.2 \text{ cm}^2

Answer

The area of the larger triangle is 147.2 cm².

Key Concept

Scale Factor in Similar Triangles

Explanation

The area of similar triangles is proportional to the square of the scale factor of their corresponding sides.

Solution by Steps

step 1

Understand that the perimeter of the larger triangle is 32 times the perimeter of one of the smaller triangles

step 2

The scale factor for the perimeter is the same as the scale factor for the sides

step 3

The number of smaller triangles that fit along one side of the larger triangle is equal to the scale factor

step 4

Calculate the scale factor for the sides:

\text{Scale factor} = \sqrt{32}

step 5

The number of smaller triangles used to form the larger triangle is the scale factor squared

step 6

Compute the number of smaller triangles:

\text{Number of smaller triangles} = (\sqrt{32})^2

step 7

Simplify to find the number of smaller triangles:

\text{Number of smaller triangles} = 32

Answer

32 smaller triangles are used to form the larger triangle.

Key Concept

Perimeter and Scale Factor in Similar Triangles

Explanation

The number of smaller triangles used to form a larger similar triangle is the square of the scale factor for the perimeters.