Question

1. solve for?

Explanation:

Step1: Apply the Angle - Side - Angle (ASA) congruence (Law of Cosines is not needed here as the two triangles share an angle and we can use the property of triangles with two sides and included angle (but here it's a theorem related to two triangles with two sides proportional and included angle equal, similar to SSS - like ratio check for congruence in a sense of split - triangle proportionality. Wait, actually, using the formula from the Angle - Bisector Theorem's converse (if \(\frac{a}{b}=\frac{c}{d}\), but here for two triangles with two sides and included angle. Wait, more accurately, using the formula for two triangles with two sides \(a_1,a_2\), \(b_1,b_2\) and included angle equal, the ratio of the third sides is same as the ratio of the other two pairs. But here, assume the two triangles (the upper and the lower) have an equal included angle. Let the unknown side be \(x\). We can use the property that if in two triangles \(\triangle_1\) and \(\triangle_2\) with sides \(a_1 = 3\), \(b_1=4\), \(a_2=x + 3\), \(b_2 = 4+8\) (no, wait, another approach. Using the formula from the theorem of two triangles with two sides and included angle. Let's use the formula: \(\frac{3}{x + 3}=\frac{4}{4 + 8}\) (by the property of triangles divided by an angle - bisector - like line, but more precisely, using the ratio of sides in two triangles with equal included angle. Cross - multiply:

\(4(x + 3)=3\times(4 + 8)\)

Step2: Expand and solve the equation

Expand \(4(x + 3)\): \(4x+12\)
\(3\times(4 + 8)=3\times12 = 36\)
So, \(4x+12 = 36\)
Subtract 12 from both sides: \(4x=36 - 12=24\)
Divide both sides by 4: \(x=\frac{24}{4}=6\)

Answer:

\(6\)