A 5 inch by 8 inch rectangle is enlarged to a similar rectangle whose smallest side measures 9 inches. What is the length in inches of the diagonal of the enlarged rectangle?
D1/D2 = 5/9
D1 = √(5^2 + 8^2) = √89
so √89/D2 = 5/9
5D2 = 9√89
D2 = 9√89/5
notice that the diagonal increased by a factor of 9/5 or the same as the increase in the sides.
Well, first things first, let's analyze the situation here. We have a 5 inch by 8 inch rectangle that is being enlarged to a similar rectangle with a smallest side of 9 inches.
Now, when we say "similar" in geometry, it means that the corresponding angles are the same, and the corresponding sides are in proportion.
Since the smallest side of the enlarged rectangle is 9 inches, and the smallest side of the original rectangle is 5 inches, we can find the scale factor by dividing 9 by 5. This gives us a scale factor of 1.8.
To find the lengths of the other sides of the enlarged rectangle, we multiply the corresponding lengths of the original rectangle by the scale factor. So, the length of the enlarged rectangle will be 8 inches multiplied by 1.8, which gives us 14.4 inches.
Now, we can find the diagonal of the enlarged rectangle using the Pythagorean theorem. The diagonal is the hypotenuse of a right triangle formed by the two sides of the rectangle. Using the lengths we found, the diagonal can be calculated as follows:
diagonal = sqrt(14.4^2 + 9^2)
Calculating that out, we get:
diagonal ≈ sqrt(207.36 + 81)
diagonal ≈ sqrt(288.36)
diagonal ≈ 16.97 inches
So, the length of the diagonal of the enlarged rectangle is approximately 16.97 inches.
To find the length of the diagonal of the enlarged rectangle, we need to determine the scale factor of enlargement.
The scale factor can be found by dividing the length of the enlarged rectangle's smallest side by the length of the original rectangle's smallest side.
Let's calculate the scale factor first:
Scale factor = (Length of enlarged rectangle's smallest side) / (Length of original rectangle's smallest side)
= 9 inches / 5 inches
= 1.8
Now, we can find the length of the diagonal of the enlarged rectangle using the scale factor.
The length of the diagonal of the original rectangle can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
Let's calculate the length of the diagonal of the original rectangle:
Diagonal of original rectangle = sqrt((5 inches)^2 + (8 inches)^2)
= sqrt(25 square inches + 64 square inches)
= sqrt(89 square inches)
≈ 9.43 inches
Now, we can calculate the length of the diagonal of the enlarged rectangle by multiplying the scale factor with the length of the diagonal of the original rectangle:
Length of diagonal of enlarged rectangle = Scale factor * Length of diagonal of original rectangle
= 1.8 * 9.43 inches
≈ 16.97 inches
Therefore, the length in inches of the diagonal of the enlarged rectangle is approximately 16.97 inches.
To find the length of the diagonal of the enlarged rectangle, we first need to determine the dimensions of the enlarged rectangle.
The original rectangle has sides measuring 5 inches and 8 inches. Let's call these dimensions x and y, respectively.
The enlarged rectangle has a smallest side measuring 9 inches. Let's call this dimension s. Since the rectangle is similar, the corresponding side of length y of the original rectangle is also increased by the same factor.
To determine the dimensions of the enlarged rectangle, we can set up a proportion:
x/s = 5/9
Solving for x, we get:
x = (5/9) * s
Now that we know x, we can find y using the same proportion:
y/s = 8/9
Solving for y, we get:
y = (8/9) * s
Now we have the dimensions of the enlarged rectangle in terms of s.
To find the length of the diagonal of the enlarged rectangle, we can use the Pythagorean theorem, because the diagonal forms a right triangle with sides x, y, and the hypotenuse being the diagonal.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So the diagonal (d) squared is equal to x squared plus y squared:
d^2 = x^2 + y^2
Substituting the expressions for x and y, we get:
d^2 = ((5/9) * s)^2 + ((8/9) * s)^2
Expanding and simplifying, we get:
d^2 = (25/81 * s^2) + (64/81 * s^2)
Combining the terms, we have:
d^2 = (25/81 + 64/81) * s^2
d^2 = (89/81) * s^2
Since the length cannot be negative, we can drop the square root and solve for d:
d = sqrt(89/81) * s
Finally, substituting the value of s (s = 9), we can calculate the length of the diagonal:
d = sqrt(89/81) * 9
Simplifying, we get:
d ≈ 9.68 inches
Therefore, the length of the diagonal of the enlarged rectangle is approximately 9.68 inches.