Triangle A has sides of lengths #36 #, #32 #, and #24 #. Triangle B is similar to triangle A and has a side of length #4 #. What are the possible lengths of the other two sides of triangle B?

Question

Triangle A has sides of lengths #36    #, #32    #, and #24     #. Triangle B is similar to triangle A and has a side of length #4     #. What are the possible lengths of the other two sides of triangle B?

Savannah Aguilar · Accepted Answer

The other two possible sides are #color(red)(3.bar 5# and #color(blue)(2.bar 6#

We know the sides of triangle A,But we know only one side of triangle B

Consider,

We can solve for the other two sides using the ratio of the corresponding sides

Solve,

#color(red)(x#

#rarr36/4=32/x#

#rarr9=32/x#

#color(green)(rArrx=32/9=3.bar 5#

#color(blue)(y#

#rarr36/4=24/y#

#rarr9=24/y#

#color(green)(rArry=24/9=2.bar 6#

Violet Aldrich · Answer

undefined To find the possible lengths of the other two sides of triangle B, we can use the property of similar triangles. Since triangles A and B are similar, their corresponding sides are in proportion.

We can set up a proportion using the corresponding sides of the two triangles. Let's denote the lengths of the sides of triangle B as $ x $ and $ y $.

The proportion can be set up as:

$$
\frac{36}{4} = \frac{x}{y}
$$

This proportion represents the ratio of corresponding sides of the two similar triangles. We can solve for $ x $ and $ y $ by cross-multiplying:

$$
36y = 4x
$$

Now, we can express $ x $ in terms of $ y $ by dividing both sides by 4:

$$
x = \frac{36y}{4} = 9y
$$

So, one of the sides of triangle B is $ 9y $. Now, we need to find the other side, $ y $.

We can use the Pythagorean theorem to find the length of the other side of triangle A:

$$
32^2 + 24^2 = 36^2
$$

$$
1024 + 576 = 1296
$$

$$
1600 = 1296
$$

$$
\sqrt{1600} = \sqrt{1296}
$$

$$
40 = 36
$$

Since the sides of triangle A satisfy the Pythagorean theorem, triangle A is a right triangle.

Now, since triangles A and B are similar, triangle B is also a right triangle. Using the Pythagorean theorem, we can find the length of the other side of triangle B:

$$
4^2 + y^2 = (9y)^2
$$

$$
16 + y^2 = 81y^2
$$

$$
16 = 80y^2
$$

$$
y^2 = \frac{16}{80}
$$

$$
y^2 = \frac{1}{5}
$$

$$
y = \sqrt{\frac{1}{5}}
$$

So, the possible lengths of the other two sides of triangle B are $ 9\sqrt{\frac{1}{5}} $ and $ \sqrt{\frac{1}{5}} $.