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.

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.

Let area of triangle DFG = x then area of triangle ABC = x + 77

To find the area of 2 triangles

ar(DFG)/ar(ABC = (5/6)²

x/(x + 77) = 25/36

36x = 25x + (25 * 77)

36x - 25x = 25 * 77

11x = 25 * 77

x = 175

Therefore area of triangle DFG = 175

Area of triangle ABC = 175 + 77 = 252

MrRoyal MrRoyal · Answer 2 · 2021-11-29T14:04:29+01:00

Similar triangles may or may not be congruent.

The areas of triangles ABC and DFG are 252 and 175 square centimeters, respectively.

The given parameters are:

[tex]\mathbf{\triangle ABC \sim \triangle DFG}[/tex] ---- similar triangles

[tex]\mathbf{ABC : DFG =6 : 5}[/tex] -- ratio of the sides

[tex]\mathbf{Area_{ABC} = Area_{DFG} + 77}[/tex] --- the areas

So, the relationship between the areas of the triangle is the square of the ratio of the side lengths.

i.e.

[tex]\mathbf{ABC : DFG =6^2 : 5^2}[/tex]

[tex]\mathbf{ABC : DFG =36 : 25}[/tex]

Express as fraction

[tex]\mathbf{\frac{ABC }{ DFG} =\frac{36 }{ 25}}[/tex]

Let triangle DFG represent x and triangle ABC represent y.

So, we have:

[tex]\mathbf{\frac{y }{ x} =\frac{36 }{ 25}}[/tex]

and

[tex]\mathbf{y = x + 77}[/tex]

[tex]\mathbf{\frac{y }{ x} =\frac{36 }{ 25}}[/tex] becomes

[tex]\mathbf{\frac{x + 77 }{ x} =\frac{36 }{ 25}}[/tex]

Cross multiply

[tex]\mathbf{25(x + 77) = 36x}[/tex]

Expand

[tex]\mathbf{25x + 1925 = 36x}[/tex]

Collect like terms

[tex]\mathbf{36x - 25x = 1925}[/tex]

[tex]\mathbf{11x = 1925}[/tex]

Divide

[tex]\mathbf{x = 175}[/tex]

Recall that:[tex]\mathbf{y = x + 77}[/tex]

So, we have:

[tex]\mathbf{y = 175 + 77}[/tex]

[tex]\mathbf{y = 252}[/tex]

Hence, the areas of triangles ABC and DFG are 252 and 175 square centimeters, respectively.