Constructing Triangles given some parameters

Question

In my maths assignment, it says:

1. Construct a triangle with perimeter $115$ mm, altitude $70$ mm and vertical angle $45^\circ$.
2. Constructing a triangle with altitude $76$ mm and base angles $60$ and $45$ degrees.
3. Construct an isosceles triangle given $135$ mm as perimeter and altitude as $55$ mm.

I am very confused on what to do from the beginning. How do I construct such Triangles?

Answer 1

Call the triangle $ABC$, and the altitude $AD$.

(1) construct a triangle with perimeter length 115 and with the altitude length 70.

That is impossible. $ADB$ is a right-angled triangle, so $AB>AD$. Similarly, $AC>AD$. So the perimeter must be greater than $2AD$.

(2) construct a triangle with $\angle B=45\circ$, $\angle C=60\circ$ and $AD=76$.

We have $AB\sin\angle B=AD$, which gives us $AB$. Similarly $AC\sin\angle C=AD$, which gives us $AC$. Similarly, $AB\cos\angle B+AC\cos\angle C=BC$. So we have all the side lengths and constructing the triangle is trivial.

(3) construct a triangle with perimeter 135, $AD=55$ and $AB=AC$. Suppose $BC=x$. Then $AB=AC=\sqrt{55^2+\left(\frac{x}{2}\right)^2}$. So we have perimeter $135=x+2\sqrt{55^2+\left(\frac{x}{2}\right)^2}$. So we can solve for $x$ (approx 22.7) and hence get the side lengths.