Question

1. find x and y

Explanation:

Step1: Use similar - triangles property

Since the three right - angled triangles are similar, the ratios of their corresponding sides are equal. Consider the two smaller similar triangles. The ratio of the vertical sides of the two smaller similar triangles is the same as the ratio of their hypotenuses. The vertical sides of the two smaller similar triangles are 1.5 and 4 respectively. Let's first find the ratio of the vertical sides. The ratio of the vertical sides of the two similar triangles is $\frac{1.5}{4}$.

Step2: Find $x$

We know that $\frac{x}{6}=\frac{1.5}{4}$. Cross - multiply to get $4x = 1.5\times6$. Then $4x=9$, and $x=\frac{9}{4}=2.25$.

Step3: Consider the largest and the middle - sized similar triangles

The vertical sides of the largest and the middle - sized similar triangles are $(4 + 1.5+2)$ and $(4 + 1.5)$ respectively. The ratio of the vertical sides of the two similar triangles is $\frac{4 + 1.5+2}{4 + 1.5}=\frac{7.5}{5.5}=\frac{15}{11}$. The ratio of the hypotenuses of these two similar triangles is also $\frac{15}{11}$. Let the hypotenuse of the largest triangle be $6 + x + y$ and the hypotenuse of the middle - sized triangle be $6 + x$. We know $x = 2.25$, so the hypotenuse of the middle - sized triangle is $6+2.25 = 8.25$. Let's use another way. Consider the two similar right - angled triangles formed by the whole and the part. The ratio of the vertical sides of the two similar triangles with vertical sides 4 and $(4 + 1.5+2)$ gives us the ratio of the hypotenuses. The ratio of the vertical sides is $\frac{4}{4 + 1.5+2}=\frac{4}{7.5}=\frac{8}{15}$. The hypotenuse of the largest triangle is $h$. We know that $\frac{6}{h}=\frac{8}{15}$, so $h=\frac{6\times15}{8}=\frac{45}{4}=11.25$. Since $x = 2.25$ and the hypotenuse of the largest triangle is $6 + x + y$, then $11.25=6 + 2.25+y$.

Step4: Solve for $y$

Rearrange the equation $11.25=6 + 2.25+y$ to get $y=11.25-(6 + 2.25)$. So $y = 3$.

Answer:

$x = 2.25$, $y = 3$