Question

Can someone help me, please? I need to find the value of x and y rounded to the nearest tenth. Thank you!​

Answer 1

Answer:

$x\approx 24.0,\\y\approx 46.4$

Step-by-step explanation:

Let the height of the largest triangle marked be $h$. We can set up the following equation:

$\sin 30^{\circ}=\frac{h}{34},\\h=34 \sin 30^{\circ}=17$ (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 $h$. Therefore, we can use the Pythagorean theorem to solve for $x$:

$17^2+17^2=x^2\\x^2=\sqrt{17^2\cdot 2},\\x=17\sqrt{2}\approx \boxed{24.0}$ (you can also use trig or 45-45-90 triangle rules which are derived from the Pythagorean theorem)

Segment $y$ 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:

$y_{right}=\sqrt{34^2-17^2}\approx \boxed{29.4}$

Therefore, we have:

$y=17+29.4=\boxed{46.4}$