Respuesta :
Answer:
Part a) The area of the smaller triangle is [tex]16\ cm^{2}[/tex]
Part b) The area of the larger triangle is [tex]49\ cm^{2}[/tex]
Step-by-step explanation:
we know that
If two figures are similar, then the ratio of its perimeters is equal to the scale factor and the ratio of its areas is equal to the scale factor squared
step 1
Find the scale factor
Let
z-----> the scale factor
[tex]z=\frac{4}{7}[/tex] ------> ratio of corresponding perimeters
step 2
Find the area of each triangle
Let
z-----> the scale factor
x-----> the area of the smaller triangle
y-----> the area of the larger triangle
we know that
[tex]x+y=65[/tex] ------> equation A
[tex]z^{2}=\frac{x}{y}[/tex]
we have
[tex]z=\frac{4}{7}[/tex]
substitute
[tex](\frac{4}{7})^{2}=\frac{x}{y}[/tex]
[tex](\frac{16}{49})=\frac{x}{y}[/tex]
[tex]x=\frac{16}{49}y[/tex] ------> equation B
substitute equation B in equation A and solve for y
[tex]\frac{16}{49}y+y=65[/tex]
[tex]\frac{65}{49}y=65[/tex]
[tex]y=49\ cm^{2}[/tex]
Find the value of x
[tex]x=\frac{16}{49}y[/tex] ------> [tex]x=\frac{16}{49}(49)=16\ cm^{2}[/tex]
[tex]\boxed{16{\text{ c}}{{\text{m}}^2}}[/tex] and [tex]\boxed{49{\text{ c}}{{\text{m}}^2}}[/tex] are the areas of two similar triangles.
Further explanation:
Explanation:
Let us assume that area of triangle 1 as A and the area of triangle 2 as B.
The given ratio of the perimeter of the two triangles is 4:7.
If two triangles are similar to each other than the ratio of areas of triangle is equal to the ratio of the square of perimeter.
The sum of areas of triangle can be expressed as follows,
[tex]\begin{aligned} A + B = 65\\ A = 65 - B \\ \end{aligned}[/tex]
The ratio of the areas of the triangles can be calculated as,
[tex]\begin{aligned}\frac{{{\text{Area of triangle 1}}}}{{{\text{Area of triangle 1}}}} &= \frac{{{{\left( {{\text{Perimeter of triangle 1}}} \right)}^2}}}{{{{\left( {{\text{Perimeter of triangle 2}}} \right)}^2}}}\\\frac{A}{B} &= \frac{{{4^2}}}{{{7^2}}}\\\frac{A}{B} &= \frac{{16}}{{49}}\\49A &= 16B \\\end{aligned}[/tex]
The value of A is [tex]65 - B.[/tex]
[tex]\begin{aligned}49 \times \left( {65 - Y} \right) &= 16Y\\49 \times 65 - 49B &= 16B\\49 \times 65 &= 16B + 49B\\49 \times 65 &= 65B\\\frac{{49 \times 65}}{{65}} &= B\\49&= B\\\end{aligned}[/tex]
The area of second triangle is [tex]49{\text{ c}}{{\text{m}}^2}.[/tex]
The area of first triangle with area A can be calculated as follows,
[tex]\begin{aligned}A&= 65 - 49\\A&= 16{\text{ c}}{{\text{m}}^2}\\\end{aligned}[/tex]
[tex]\boxed{16{\text{ c}}{{\text{m}}^2}}[/tex] and [tex]\boxed{49{\text{ c}}{{\text{m}}^2}}[/tex] are the areas of two similar triangles.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Triangles
Keywords: perimeter, ratio, similarity, similar, triangles, proportional, square, area, area of triangle, two similar triangles, corresponding sides, sum of areas, 65 cm2, 4:7.
