# Constructing Triangles given some parameters

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?

• The second looks like the easiest. Can you see how to get the sides? [Do you know some basic formulae like opposite/hypotenuse = sine?] Jan 25, 2020 at 12:01
• I know how, but the angles and altitude are giving me headaches... Jan 25, 2020 at 12:08
• If I use 45°for b, 60° will not fit, because of the altitude Jan 25, 2020 at 12:10
• SOHCAHTOA right? Jan 25, 2020 at 12:12
• In (1) the perimeter is larger than twice the altitude, when this is inside the triangle. Since 115<2\cdot 70 it must be that the altitude is outside. Jan 25, 2020 at 12:25

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.

• Any pictures or videos on how I can construct this? Jan 25, 2020 at 17:22
• Should I use the perimeter as the base line? Jan 25, 2020 at 17:23
• @YNK Thanks. Fixed. Jan 25, 2020 at 18:22