Question

Each dimension of a right triangle with legs of length 6 cm and 8 cm and a hypotenuse of length 10 cm is multiplied by 1/2 to form a similar right triangle. How is the ratio of perimeters related to the ratio of corresponding sides? How is the ratio of areas related to the ratio of corresponding sides?

Answer 1

Answer:

Ratio of the perimeters is equal to the ratio of the corresponding sides.

Ratio of areas is equal to square of the ratio of the corresponding sides.

Step-by-step explanation:

Two triangles are said to be similar if they have same shape but not necessarily same size.

Perimeter of triangle refers to sum of all sides .

Sides of triangle are of length, 8 cm,  6 cm , 10 cm

So, perimeter of original triangle: 6+8+10=24 cm

Sides of the other triangle:

$\frac{6}{2}=3\,,\,\frac{8}{2}=4\,,\frac{10}{2}=5$

Perimeter of new triangle: 3+4+5=12 cm (You get 3, 4, and 5 from dividing 6, 8. and 10 by 2.)

Ratio of original triangle to new triangle: $\frac{24}{12}=2$

Also, ratio of sides of original triangle to new triangle: $\frac{6}{3}=\frac{8}{4}=\frac{10}{5}=2$

The ratio of the perimeters is equal to the ratio of the corresponding sides, as the original measurements are two times the length of the new measurements.

We know that area of right angles triangle = $\frac{1}{2}\times base\times height$

Area of original triangle: (6x8)/2=24 cm^2

Area of new triangle: (3x4)/2=6 cm^2

Ratio of areas of original triangle to new triangle = $\frac{24}{6}=4=2^2$

So, ratio of areas is equal to square of the corresponding sides.

Answer 2

Perimeter of original triangle: 6+8+10=24 cm
Perimeter of new triangle: 3+4+5=12 cm (You get 3, 4, and 5 from dividing 6, 8. and 10 by 2.)
Ratio of original to new is 24 to 12, simplified to 2 to 1. 
The ratio of the perimeter is the ratio of the corresponding sides, as the original measurements are two times the length of the new measurements.

Area of original triangle: (6x8)/2=24 cm^2
Area of new triangle: (3x4)/2=6 cm^2
Ratio of original to new is 24 to 6, simplified to 4 to 1.