Question

The ratio of the perimeters of two similar triangles is 4:7. What is the area of each of these triangles if the sum of their areas is 65cm^2?

Answer 1

The area of the larger triangle is $\boxed{49{\text{ c}}{{\text{m}}^2}}$ and the area of the smaller triangle is $\boxed{16{\text{ c}}{{\text{m}}^2}}.$

Further explanation:

Given:

The ratio of the perimeters of two similar triangles is $4:7.$

Explanation:

If triangles are similar then the ratio of the areas is equal to the square ratio of corresponding sides as well as the square of the perimeter.

Assume that the area of larger triangle as Q.

Assume that the area of smaller triangle as P.

The given ratio of the perimeters of the triangle is $4:7.$

The ratio of the areas of the larger and smaller triangle can be expressed as follows,

$\begin{aligned}\frac{{{\text{Area of smaller triangle}}}}{{{\text{Area of larger triangle}}}} &= \frac{{{{\left( {{\text{Perimeter of smaller triangle}}} \right)}^2}}}{{{{\left( {{\text{Perimeter of larger triangle}}} \right)}^2}}}\\\frac{{\text{P}}}{{\text{Q}}} &= \frac{{{4^2}}}{{{7^2}}}\\\frac{{\text{P}}}{{\text{Q}}} &= \frac{{16}}{{49}} \\ {\text{49P}} &= {\text{16Q}}\\\end{aligned}$

The sum of area of larger triangle and the area of smaller triangle is $65{\text{ c}}{{\text{m}}^2}.$

$\begin{aligned}{\text{P}} + {\text{Q}}&= 65\\{\text{Q}}& = 65 - {\text{P}}\\\end{aligned}$

Now, substitute $65 - P$ for $Q$ in equation $49P = 16Q.$

$\begin{aligned}49{\text{P}} &= 16{\text{Q}}\\49{\text{P}} &= 16 \times \left( {65 - {\text{P}}} \right)\\49{\text{P}} &= 16 \times 65 - 16{\text{P}}\\49{\text{P}} + 16{\text{P}} &= 16 \times 65\\65{\text{P}} &= 16 \times 65\\{\text{P}} &= \frac{{16 \times 65}}{{65}}\\{\text{P}} &= 16{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of larger triangle can be calculated as follows,

$\begin{aligned}{\text{Q}} &= 65 - {\text{P}}\\&= 65 - 16\\&= 49{\text{ c}}{{\text{m}}^2}\\\end{aligned}$

The area of the larger triangle is $\boxed{49{\text{ c}}{{\text{m}}^2}}$ and the area of the smaller triangle is $\boxed{16{\text{ c}}{{\text{m}}^2}}.$

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Triangles

Keywords: Ratio, perimeter, similar, similarity, triangles, proportional, square, area, area of triangle, two similar triangles, sum, sum of areas, 65 cm2, 4:7, corresponding sides.

Answer 2

Answer:

  16 cm^2 and 49 cm^2

Step-by-step explanation:

The ratio of areas is the square of the ratio of linear dimensions, so is ...

  (4/7)^2 = 16/49

The sum of these "ratio units" is 16 +49 = 65. The sum of triangle areas is 65 cm^2, so each "ratio unit" must stand for 1 cm^2 of area. That is, the triangles must have areas of ...

• 16 cm^2
• 49 cm^2