I'm at a loss for 7 b. given I am not allowed to use trigonometry.

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h ~ 7.5 cm ?

In the first case, the base length is 12.3 cm and altitude is 5.6 cm. We can easily compute the area of the triangle.

Area (A) = 1/2 × base × height

Or 1/2 × 12.3 × 5.6

= ~ 34.4 square centimeters

In the second case,

h is the altitude drawn from the side of length 9.2 cm to 12.3 cm. This implies that the base of the triangle in this case is 9.2 cm and altitude = h cm.

=> A = 1/2 × 9.2 × h

Area in case 1 = Area in case 2 (as we are considering the same triangle)

=> 34.4 = 4.6h or h ~ 7.5 cm.
A Year 10 student who is a friend of the family asked me about this question. They have not yet covered trigonometry, so that cannot be used in the answer.

I have Googled and Googled and cannot think of a good way to use the answer from (a) to solve (b), given that it is a different triangle. Does anyone have any insight to offer?
by
(2 * S)/9.2?
Because
S=(9.2 * h)/2
Triangles area = 1/2. * Height * base

Here base is 9.2 as you are projecting height onto that side in the picture
7a is a direct answer (1/2 *bh)

Now for 7b,

From the given qualities, we have the two sides and height of the triangle. Now the larger triangle can be split into two smaller ones, from which we can then obtain the length of the side of the smaller triangle. This can then be used to evaluate the area of the smaller triangle.

As for the dotted triangle, it's area should be equal to the area of the before mentioned smaller triangle (that's my best guess). This is because, both of it is right triangles with a common hypotenuse. Now, we can apply 1/2 * bh in reverse to get h

This is my best guess, it might be wrong.
Trigonometry is literally the study of triangles.  So you can declare the problem impossible by definition to do without trigonometry.
(1/2)bh=(1/2)(12.3)(5.6)=34.4
1/2 * 12.3 * 5.6 = 1/2 * 9.2h //
68.88 = 9.2h //
h = 7.49 (3sf) //