# 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?

The other two possible sides are

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,

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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}} ).

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