Triangles A B C and D E F are similar triangles.
Two similar triangles.Short description, Two similar triangles.,Long description,
Triangle A B C and triangle D E F are similar.
Side A B is labeled 9 and three-fourths cm and is similar to side D E that is labeled 3 and one-fourth cm.
Side B C is labeled 18 and three-fourths cm and is similar to side E F that is labeled n.
Side C A is labeled 27 and three-fourths cm and is similar to side F D that is labeled 9 and one-fourth cm.
Question
What is the length of side n?
To find the length of side n, we can set up a proportion using the corresponding sides of the similar triangles:
AB/DE = BC/EF = CA/FD
Substituting the given values:
9 and 3/4 cm / 3 and 1/4 cm = 18 and 3/4 cm / n = 27 and 3/4 cm / 9 and 1/4 cm
To simplify the first ratio, we can convert the mixed numbers to improper fractions:
39/4 cm / 13/4 cm = 18 and 3/4 cm / n = 111/4 cm / 37/4 cm
Now we can set up the proportion:
(39/4 cm / 13/4 cm) = (18 and 3/4 cm / n) = (111/4 cm / 37/4 cm)
Cross-multiplying gives us:
(39/4 cm) * n = (13/4 cm) * (18 and 3/4 cm)
(18 and 3/4 cm) * (37/4 cm) = (13/4 cm) * (111/4 cm)
Now we can simplify the equation:
39n/4 = (243/4) + (27/4)
(37/4) * n = (351/4) + (81/4)
Combining like terms:
39n/4 = 324/4 + 81/4
(37/4) * n = (405/4)
Now, we can solve for n by multiplying both sides by 4/37:
n = (405/4) * (4/37)
n = (405/37)
Therefore, the length of side n is 405/37 cm.
To find the length of side n, we can use the concept of similarity. Since the triangles A B C and D E F are similar, the corresponding sides are proportional.
The ratio of corresponding sides A B to D E is 9 and three-fourths cm to 3 and one-fourth cm. So we can set up the following proportion:
(9 and three-fourths cm) / (3 and one-fourth cm) = (18 and three-fourths cm) / n
To solve for n, we can cross-multiply and solve for n:
(9 and three-fourths cm) * n = (3 and one-fourth cm) * (18 and three-fourths cm)
To simplify the equation, we can convert the mixed numbers to improper fractions:
(9 and three-fourths cm) * n = (3 and one-fourth cm) * (18 and three-fourths cm)
= (39/4 cm) * n = (13/4 cm) * (75/4 cm)
Now we can multiply the fractions:
(39/4 cm) * n = (13/4 cm) * (75/4 cm)
= (39/4 cm) * n = (13 * 75) / (4 * 4) cm
Next, we can simplify the expression:
(39/4 cm) * n = (975/16) cm
To solve for n, we can divide both sides by (39/4 cm):
n = (975/16) cm / (39/4 cm)
Simplifying further:
n = (975/16) cm * (4/39 cm)
= 975/16 * 4/39 cm
Finally, we can multiply the fractions:
n = (975 * 4) / (16 * 39) cm
= 3900/624 cm
The length of side n is 6 and one-fourth cm.
To find the length of side n, we can use the principle of similar triangles which states that corresponding sides of similar triangles are proportional.
Let's set up a proportion using the corresponding sides of the triangles:
AB/DE = BC/EF = CA/FD
Substituting the given values:
9 and three-fourths cm / 3 and one-fourth cm = 18 and three-fourths cm / n = 27 and three-fourths cm / 9 and one-fourth cm
Now, we can solve for the unknown side, which is represented by n.
To solve for n, we can cross multiply the first and second fractions:
(9 and three-fourths cm) * n = (3 and one-fourth cm) * (18 and three-fourths cm)
To simplify the right side of the equation:
(9 and three-fourths cm) * n = (13/4 cm) * (75/4 cm)
Next, multiply the fractions on the right side:
(9 and three-fourths cm) * n = (975/16 cm^2)
To isolate n, divide both sides by (9 and three-fourths cm):
n = (975/16 cm^2) / (9 and three-fourths cm)
Convert the mixed fraction (9 and three-fourths cm) to an improper fraction:
n = (975/16 cm^2) / (39/4 cm)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
n = (975/16 cm^2) * (4/39 cm)
Multiply the numerators and the denominators:
n = (975 * 4) / (16 * 39 cm)
Simplifying the multiplication:
n = 3900 / 624 cm
Now, divide the numerator by the denominator:
n = 6 and one-fourth cm
Therefore, the length of side n is 6 and one-fourth cm.