Triangle A has sides of lengths #24 #, #15 #, and #21 #. Triangle B is similar to triangle A and has a side of length #24 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #24    #, #15    #, and #21     #. Triangle B is similar to triangle A and has a side of length #24     #. What are the possible lengths of the other two sides of triangle B?

Sadie Adams · Accepted Answer

Case 1 : #color(green)(24, 15,21# Both are identical triangles

Case 2 : #color(blue)(24, 38.4, 33.6#

Case 3 : #color(red)(24, 27.4286, 17.1429#

Given :Triangle A (#DeltaPQR#) similar to Triangle B #(DeltaXYZ)#

#PQ = r = 24, QR = p = 15, RP = q = 21#

Case 1 : #XY = z = 24#

Then using similar triangles property,

#r / z = p / x = q / y#

#24 / 24 = 15 / x = 21 / y#

#:. x = 15, y = 21#

Case 2 : #YZ = x = 24#

#24 / z = 15 / 24 = 21 / y#

#z = (24 * 24) / 15 = 38.4#

#y = (21 * 24) / 15 = 33.6#

Case 2 : #ZX = y = 24#

#24 / z = 15 / x = 21 / 24#

#z = (24 * 24) / 21 = 27.4286#

#y = (15 * 24) / 21 = 17.1429#

Noah Atkinson · Answer

undefined To find the possible lengths of the other two sides of Triangle B, we need to use the concept of similarity between triangles. Since Triangle B is similar to Triangle A, their corresponding sides are in proportion.

Let's denote the lengths of the sides of Triangle B as x and y. According to the given information, the side lengths of Triangle A are 24, 15, and 21, and the corresponding side of Triangle B is 24.

Using the property of similarity, we can set up the proportion:
$ \frac{x}{24} = \frac{15}{24} $
$ \frac{y}{24} = \frac{21}{24} $

Solving for x and y:
$ x = \frac{15}{24} \times 24 $
$ y = \frac{21}{24} \times 24 $

$ x = 15 $ and $ y = 21 $

Therefore, the possible lengths of the other two sides of Triangle B are 15 and 21.