Question

Two corresponding sides of similar triangles are 2 cm and 5 cm. What is the area of the second triangle if the area of the first triangle is 8 cm2?

The triangles ABC and DFG are similar and the ratio of their corresponding sides is 6:5. The area of the triangle ABC is greater than the area of the triangle DFG by 77 cm2. Find the areas of these triangles.

Answer 1

The second triangle's area is 50 cm². The ratios of the sides of triangles ABC and DFG create an area ratio of 36:25, which helps to solve for their individual areas. For similar square figures, the area ratio is the scale factor squared.

Step-by-step explanation:

To find the area of the second triangle when two corresponding sides of similar triangles are given, we use the scale factor, which in this case is ²/₅. Since areas of similar figures scale with the square of the scale factor, the area of the second triangle is (5/2)2 × 8 cm2 = 25/4 × 8 cm2 = 50 cm2.

For the triangles ABC and DFG with corresponding sides in the ratio 6:5, and given that the area of ABC is greater than that of DFG by 77 cm2, we use the areas' ratio, which is the square of the sides ratio, to find each area. If the area of DFG is x cm2, then the area of ABC is x + 77 cm2. The ratio of their areas is (x + 77)/x = (6/5)2. Solving for x yields the area of DFG and subsequently that of ABC.

When comparing the areas of similar figures, like the squares in Marta's example, the ratio of areas is the square of the scale factor. Since Marta's second square has sides twice as long, the area ratio is 22:1 = 4:1, meaning the area of the larger square is 4 times that of the smaller square.

Answer 2

(1) Area of second triangle= 50 cm^2.

(2) Area of ΔABC=98 cm^2.

and Area of ΔDFG=175 cm^2.

Step-by-step explanation:

" For two similar triangles the ratio of sides is equal to the ratio of square of their areas".

i.e. if a,b are the corresponding sides of two similar triangles and let A and B denote the area of two triangles then we have the relation as:

`(a^2)/(b^2)=(A)/(B)`

(1)

for the first question:

we have a=2, b=5.

A=8 cm^2.

Hence,

`(2^2)/(5^2)=(8)/(B)\\\\(4)/(25)=(8)/(B)`

B=50 cm^2.

Hence, the area of second triangle is 50 cm^2.

(2)

In second option we have:

a=6 and b=5.

A-B=77 cm^2.

A=77+B

`(6^2)/(5^2)=(A)/(B)\\  \\(36)/(25)=(77+B)/(B)\\  \\36B=25*77+25B\\\\36B-25B=25*77\\\\11B=25*77\\\\B=25*7\\\\B=175`

Hence area of second triangle i.e. ΔDFG is 175 cm^2.

and area of first triangle i.e. ΔABC=175-77=98 cm^2.