How can I find the height of $ABCD?$

Question

I found these values:

$$\frac{ah}{2}=14$$

$$\frac {a+b}{2}×h=23$$

Finally, I found $ah=28$, $bh=18$, $a=\frac{14b}{9}$.

Here, $a//b$ and $S$ is area.

Now, what should I do?

Answer 1

It is correct, all what you can find by the givens is that

$$h=\frac{46}{a+b}$$

and also that

$$\frac{ah}{2}=14 \quad \frac{bh}{2}=9 \implies \frac ab=\frac{14}{9}$$

Since we have three unknowns and 2 equations to solve completely we need some other infomation.

Answer 2

I think some information is missing in this question. What you have done up to there is completely fine but you need one more equation to solve the system. At this point, we can have $a = 14$, $b = 9$ and $h = 2$. But we also can have $a = 28$, $b = 18$ and $h = 1$. So we can't have an exact value for $h$ with the given information.

However, if question asks $h$ with respect to $a$ and $b$, as you have already found, it is $$h = \frac{46}{a+b}$$