Question

1. find a.

Explanation:

Step1: Use the mid - segment theorem

If a line segment joins the mid - points of two sides of a triangle, it is parallel to the third side and half its length. Assume the line segment with endpoints on the two sides of the triangle is a mid - segment. Let's assume the two segments on the sides of the triangle are mid - segments. We can set up a proportion. Let the two similar triangles be considered. The ratio of the corresponding sides of similar triangles is equal.
Let the two triangles be $\triangle NXY$ and $\triangle NOD$. Since the segments are mid - segments, we know that $\frac{NX}{NO}=\frac{XY}{OD}$.
We know that $OD = 4 + 12=16$ and assume the ratio of similarity is based on the mid - segment property.
Let's assume the smaller triangle and the larger triangle are in the ratio of $1:2$ (if the line is a mid - segment). But we can also use the property of similar triangles.
We have $\frac{a}{19 - a}=\frac{4}{12}$ (assuming the segments divide the sides proportionally).
Cross - multiply:
$12a=4\times(19 - a)$

Step2: Expand and solve the equation

Expand the right - hand side: $12a = 76-4a$.
Add $4a$ to both sides: $12a + 4a=76$, so $16a = 76$.
Divide both sides by 16: $a=\frac{76}{16}=\frac{19}{4} = 4.75$

Answer:

$a = 4.75$