# In the figure given identify the congruent and/or similar triangles and find the value of x and y?

Mar 17, 2018

See explanation.

#### Explanation:

From the information about side lengths we can see that all angles are right angled and isosceles. we can say that:

• $D E F$ and $B S T$ are similar to $D B C$ (all 3 triangles have identical angles).

$| E F | = 2 | D B |$, so the scale of simmilarity is $k = 2$

Now we know that: $| D F | = 2 | D C |$. Since $| D F | + | D C | = 3 x$, we have: $| D F | = x$ and $| C D | = | B D | = 2 x$

• $D E F$ is congruent to $B S T$ (corresponding sides are equal)

To calculate $x$ we use the Pythagorean theorem for triangle $B C D$

## $| B D {|}^{2} + | C D {|}^{2} = | B C {|}^{2}$

${\left(2 x\right)}^{2} + {\left(2 x\right)}^{2} = {\left(4 \sqrt{18}\right)}^{2}$

$4 {x}^{2} + 4 {x}^{2} = 16 \cdot 18$

$8 {x}^{2} = 16 \cdot 18$

${x}^{2} = 36 \implies x = 6$

Now w euse the Pythagorean theorem for $D E F$ to calculate $y$

${6}^{2} + {6}^{2} = {y}^{2}$

${y}^{2} = 72 \implies y = 6 \sqrt{2}$