Answer:
[tex]x\approx 24.0,\\y\approx 46.4[/tex]
Step-by-step explanation:
Let the height of the largest triangle marked be [tex]h[/tex]. We can set up the following equation:
[tex]\sin 30^{\circ}=\frac{h}{34},\\h=34 \sin 30^{\circ}=17[/tex] (if you're unfamiliar with trig, this is likely introduced to you as 30-60-90 triangle rules)
This height is also a leg of a 45-45-90 triangle, as marked in the diagram. From the isosceles-base-theorem, the other leg of this triangle must also be equal to [tex]h[/tex]. Therefore, we can use the Pythagorean theorem to solve for [tex]x[/tex]:
[tex]17^2+17^2=x^2\\x^2=\sqrt{17^2\cdot 2},\\x=17\sqrt{2}\approx \boxed{24.0}[/tex] (you can also use trig or 45-45-90 triangle rules which are derived from the Pythagorean theorem)
Segment [tex]y[/tex] consists of two shorter segments, a left segment and a right segment. We've already found that the left segment is equal to 17. To find the right segment we can use trig, the Pythagorean theorem, or 30-60-90 triangle rules (derived from the Pythagorean theorem):
Using Pythagorean Theorem:
[tex]y_{right}=\sqrt{34^2-17^2}\approx \boxed{29.4}[/tex]
Therefore, we have:
[tex]y=17+29.4=\boxed{46.4}[/tex]
