Question

1. what is the ratio of the sides for triangle hil?

(image of a geometric figure with points h, i, j, l, k; right angles at i (between h, i, l) and at j (between j, k, and the other leg); ij = 11 cm, jk = 22 cm; angle at h is 30°, angle at j (in triangle ijk) is 45°; angles y° at l (between i, l, j) and x° at k (between l, k, j))

Explanation:

Step1: Identify Triangle Type

Triangle HIL is a right - triangle (since $\angle HIL = 90^{\circ}$) with $\angle H=30^{\circ}$. In a $30 - 60 - 90$ right - triangle, the ratio of the sides opposite to $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$ angles is $1:\sqrt{3}:2$.
First, we can find the length of IL. Looking at triangle JKL (or the right - triangle with $45^{\circ}$), but also, we can see that the vertical side (JK) is 22 cm, and since triangle IJL and other triangles might be related, but for triangle HIL, we know that in a $30 - 60 - 90$ triangle, if we consider the side opposite $30^{\circ}$ (IL), the side opposite $60^{\circ}$ (HI), and the hypotenuse (HL).
We can also note that IL is equal to JK? Wait, no. Wait, triangle HIL: $\angle H = 30^{\circ}$, right - angled at I. So the sides: let's denote the side opposite $30^{\circ}$ (IL) as $a$, the side opposite $60^{\circ}$ (HI) as $a\sqrt{3}$, and the hypotenuse (HL) as $2a$.
From the diagram, we can see that the length of IL can be found from the other right - triangle. Wait, the triangle with $45^{\circ}$: triangle JKL (right - angled) with $\angle J = 45^{\circ}$, so it's an isosceles right - triangle. So IJ = 11 cm, and the vertical side (let's say the side from J to the bottom) is 22 cm? Wait, no, maybe IL is equal to the height. Wait, actually, in triangle HIL, since $\angle H = 30^{\circ}$ and it's a right - triangle, the ratio of the sides (opposite $30^{\circ}$ : opposite $60^{\circ}$ : hypotenuse) is $1:\sqrt{3}:2$.

Step2: Confirm the Ratio

In a $30 - 60 - 90$ triangle, the ratio of the sides is always $1:\sqrt{3}:2$. This is a standard result in geometry for special right - triangles. The side opposite the $30^{\circ}$ angle is the shortest side, the side opposite $60^{\circ}$ is $\sqrt{3}$ times the shortest side, and the hypotenuse is twice the shortest side.

Answer:

The ratio of the sides of triangle HIL (a $30 - 60 - 90$ triangle) is $1:\sqrt{3}:2$ (or in the order of the sides opposite $30^{\circ}$, $60^{\circ}$, $90^{\circ}$ angles). If we consider the sides as IL (opposite $30^{\circ}$), HI (opposite $60^{\circ}$), and HL (hypotenuse), the ratio is $IL:HI:HL = 1:\sqrt{3}:2$.