We are given three similar triangles. The sides of the largest triangle are 25, 15, and 20. We need to find the area of the smallest triangle.

1. Problem Description

We are given three similar triangles. The sides of the largest triangle are 25, 15, and

2

0. We need to find the area of the smallest triangle.

2. Solution Steps

First, we verify that the given triangle with sides 25, 15, and 20 is a right triangle. We check if the Pythagorean theorem holds.

$25^2 = 625$

$15^2 + 20^2 = 225 + 400 = 625$

Since $25^2 = 15^2 + 20^2$, the triangle is a right triangle.

The area of the large triangle is:

$A_{large} = \frac{1}{2} \cdot 15 \cdot 20 = 150$

The three triangles are similar. The smallest triangle has sides $y$, $z$, and a side that is perpendicular to the hypotenuse. We denote the side lengths of the middle triangle as $x,z,20$. We denote the side lengths of the small triangle as $y,z,15$.

The similar triangles are:

Triangle 1 (Large): Sides are 25, 15,

2

0. Area is

1

5

0. Triangle 2 (Middle): Sides are x, z,

2

0. This one has a hypotenuse of length

2

0. Triangle 3 (Small): Sides are y, z,

1

5. This one has a hypotenuse of length

1

5.

Since the triangles are similar, the ratios of corresponding sides are equal.

$\frac{x}{25} = \frac{z}{15} = \frac{20}{20/x}$ and $\frac{y}{25} = \frac{z}{20} = \frac{15}{25}$

From $\frac{y}{25} = \frac{15}{25}$, we have $y = 15$. This is obviously incorrect.

We can use the area to derive $z$.

The area of a triangle is given by $\frac{1}{2} \cdot base \cdot height$. We also have the formula for similar triangles that Area_small/Area_large = (ratio_of_sides)^2

Since the three triangles are similar right triangles, let's focus on the triangle with sides 15, 20,

2

5. The smallest triangle is similar to the original triangle. The legs of the original triangle are 15 and

2

0. We can say that $x+y = 25$. The height of the large triangle is also the height of the perpendicular line that creates the other two smaller triangles.

We calculate the height by Area = $\frac{1}{2} \cdot base \cdot height$.

$150 = \frac{1}{2} \cdot 25 \cdot h$

$300 = 25h$

$h = \frac{300}{25} = 12$

Since the triangles are similar, the sides of the smallest triangle are $y$, $z$, and $h=12$. And the hypotenuse for the small triangle is of length 15, and the base of the length is 15, and height h, where h = $\sqrt(15^2 - x^2)$. Also the other triangle is length

2

0. The similarity factor is the length of the similar triangle to the overall larger triangle.

Ratio = 15/25 = 3/5

Then, area = $150*(3/5)^2= 150 * 9/25 = 6 * 9 = 54$.

Alternatively, we have $y/25 = 3/5$ then $y = 15$,

$Area = 1/2 y h$

The ratio of the areas is $(\frac{15}{25})^2 = (\frac{3}{5})^2 = \frac{9}{25}$

Area of smallest triangle = $\frac{9}{25} \times 150 = 9 \times 6 = 54$

3. Final Answer

The area of the smallest triangle is 54 square inches.