Question

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

Answer 1

See explanation.

Explanation:

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

• `DEF` and `BST` are similar to `DBC` (all 3 triangles have identical angles).

`|EF|=2|DB|`, so the scale of simmilarity is `k=2`

Now we know that: `|DF|=2|DC|`. Since `|DF|+|DC|=3x`, we have: `|DF|=x` and `|CD|=|BD|=2x`

• `DEF` is congruent to `BST` (corresponding sides are equal)

To calculate `x` we use the Pythagorean theorem for triangle `BCD`

`|BD|^2+|CD|^2=|BC|^2`

`(2x)^2+ (2x)^2=(4sqrt(18))^2`

`4x^2+4x^2=16*18`

`8x^2=16*18`

`x^2=36=> x=6`

Now w euse the Pythagorean theorem for `DEF` to calculate `y`

`6^2+6^2=y^2`

`y^2=72=>y=6sqrt(2)`