Triangle A has sides of lengths #24 #, #15 #, and #18 #. 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 #18     #. 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?

Autumn Armstrong · Accepted Answer

Possibility 1: 15 and 18
Possibility 2: 20 and 32
Possibility 3: 38.4 and 28.8

First we define what a similar triangle is. A similar triangle is one in which either the corresponding angles are the same, or the corresponding sides are the same or in proportion. In the 1st possibility, we assume that the length of the sides of triangle #B# didn't change, so the original lengths are kept, 15 and 18, keeping the triangle in proportion and thus similar. In the 2nd possibility, we assume that the length of one side of triangle #A#, in this case length 18, has been multiplied up to 24. To find the rest of the values, we first divide #24/18# to get #1 1/3 #. Next, we multiply both #24 * 1 1/3# and #15 * 1 1/3#, and we do this to keep the triangle in proportion and thus similar. So, we get the answers of 20 and 32 In the 3rd possibility we do the exact same thing, except using the number 15. So we divide #24/15 = 1.6#, multiply #24 * 1.6# and #18 * 1.6# to get 38.4 and 28.8. Again, this is done to keep the sides in proportion, and thus the triangle is made similar.

Aurora Ayala · Answer

undefined To find the possible lengths of the other two sides of Triangle B, which is similar to Triangle A, we can use the properties of similar triangles. Since the corresponding sides of similar triangles are proportional, we can set up a proportion based on the ratios of the corresponding sides of Triangles A and B.

Let x and y represent the lengths of the other two sides of Triangle B. Then, we have the following proportion:

$$ \frac{24}{x} = \frac{15}{y} = \frac{18}{24} $$

We can solve this proportion to find the possible lengths of x and y. First, solve for x:

$$ \frac{24}{x} = \frac{18}{24} $$
$$ 24 \times 24 = 18x $$
$$ 576 = 18x $$
$$ x = \frac{576}{18} $$
$$ x = 32 $$

Now, solve for y:

$$ \frac{15}{y} = \frac{18}{24} $$
$$ 15 \times 24 = 18y $$
$$ 360 = 18y $$
$$ y = \frac{360}{18} $$
$$ y = 20 $$

Therefore, the possible lengths of the other two sides of Triangle B are 32 and 20.