Question

What is YZ? Enter your answer as a decimal in the box.

Answer 1

Answer:

YZ = 18.4 in

Step-by-step explanation:

The midsegment YZ is half the length of the third side VX , then

YZ = $\frac{1}{2}$ × 36.4 = 18.2

Answer 2

Answer:

$YZ = 18.4$

Step-by-step explanation:

1. Approach

The easiest way to solve this problem is via the method of similar triangles. One can prove that the triangle (XWV) and (ZWY) are similar. After doing so, one can find the ratio of similitude between the triangles. Finally, one can use the ratio of similitude to find the length of the segment (YZ).

2. Prove (XWV) is similar to (ZWY)

If two triangles are similar, then they are scaled versions of each other. This essentially means that the triangles have congruent angles between them, and to convert from one triangle to another, one must multiply all of the sides by a certain factor. One method that can be used to prove this is the (side-angle-side) method, also named the (SAS) theorem.

It is given that;

(WZ) = (ZX)

(WZ) + (ZX) = (WX)

Therefore, the following statement can be made;

2(WZ) = (WX)

Moreover, it is also given that,

(WY) + (YV)

(WY) + (YV) + (YV)

Therefore, one can state the following,

2(WY) = WV

Both triangles contain angle (W), therefore, by the reflexive property one can state the following,

(<W) = (<W)

Therefore by (SAS) the triangle (XWV) and (ZWY) are similar.

3. Find the ratio of similitude

The ratio of similitude is the scaling factor between two similar triangles. One multiplies the sides of one of the triangles by this number to find the side lengths of the other triangle. Since one has proven that the triangles (XWV) and (ZWY) are similar, one has to divide the corresponding sides by each other to find the ratio of similtude.

Sides (WY) and (WV) are similar as per the given information. As states above (2(WY) = WV). Divide these two sides to find the ratio of similtude,

$\frac{WY}{WV}=\frac{WY}{2(WY)}=\frac{1}{2}$

The ratio of similitude is the following, ($\frac{1}{2}$).

4. Find the length of (ZY)

To change from the sides of the triangle (XWV) to the sides of (ZWY) multiply the corresponding sides by the ratio of similitude.

$(XV) * \frac{1}{2}= (ZY)\\\\36.8 * \frac{1}{2} = (ZY)\\\\18.4 = (ZY)$