Question

1. what is the length of segment hk?

(image of a geometric figure with points h, i, j, l, k, angles 30°, 45°, right angles, and lengths 11 cm, 22 cm)

Explanation:

Step1: Analyze Triangle IJL

Triangle IJL is a right triangle with a 45° angle, so it's an isosceles right triangle. Thus, \( IJ = IL = 11\) cm? Wait, no, wait. Wait, the right angle at I and J? Wait, the right angle at I (between H and I and I and L) and right angle at J (between I and J and J and K). Wait, triangle JKL? Wait, no, first, let's look at the right triangle with 45°, which is triangle JIL? Wait, IJ is 11 cm, angle at J is 45°, right angle at I? Wait, no, the right angle is at the red square, so triangle IJL: right angle at I, angle at J is 45°, so angle at L (y°) is 45°, so it's isosceles, so \( IL = IJ = 11\) cm? Wait, no, IJ is 11 cm, so IL should be equal to IJ? Wait, no, in a right isosceles triangle, the legs are equal. So if angle at J is 45°, right angle at I, then legs IJ and IL are equal? Wait, IJ is 11 cm, so IL = 11 cm? Wait, but then the vertical side JK is 22 cm. Wait, maybe another approach. Wait, the triangle with 30°: triangle HKL? Wait, no, triangle HIL? Wait, no, let's see the length of JL. Wait, maybe first find JL. Wait, in triangle JKL? No, let's look at the right triangle with 45°: triangle JIL (right at I, angle 45° at J), so \( IL = IJ \times \tan(45°) = 11 \times 1 = 11\) cm? No, that's not right. Wait, maybe the height? Wait, no, the vertical segment JK is 22 cm. Wait, maybe the triangle with 30°: in triangle HKL, angle at H is 30°, so the side opposite 30° is half the hypotenuse. Wait, but first, we need to find the length of HL or KL? Wait, no, the segment HK: let's see the horizontal line HIJ, and the vertical line JK is 22 cm. Wait, maybe the length from H to K: first, find the length of HL. Wait, no, let's look at the right triangle with 30°: angle at H is 30°, right angle at I (between H and I and I and L). Wait, IL is equal to JK? Wait, JK is 22 cm. Wait, maybe IL is 22 cm? Wait, no, the diagram shows JK is 22 cm. Wait, maybe I made a mistake. Wait, let's re-examine. The right triangle with 45°: triangle JIL (right at I, angle 45° at J), so \( JL = IJ / \cos(45°) = 11 / (\sqrt{2}/2) = 11\sqrt{2}\), but that doesn't seem right. Wait, no, maybe the triangle with 30°: angle at H is 30°, so the side opposite 30° is half the hypotenuse. Wait, if the opposite side is 22 cm (since JK is 22 cm, and maybe IL is 22 cm), then the hypotenuse HK would be \( 2 \times 22 = 44\) cm? Wait, because in a 30-60-90 triangle, the side opposite 30° is half the hypotenuse. So if angle at H is 30°, and the side opposite (which is KL or IL) is 22 cm, then HK (the hypotenuse) is \( 2 \times 22 = 44\) cm. Let's check: in triangle HIL (right at I), angle at H is 30°, so \( \sin(30°) = \text{opposite}/\text{hypotenuse} = IL / HK \). If IL is 22 cm (since JK is 22 cm, and IL is equal to JK? Because the figure looks like IL and JK are both vertical segments), then \( \sin(30°) = 22 / HK \), so \( HK = 22 / 0.5 = 44\) cm. That makes sense. So step by step:

Step1: Identify the 30-60-90 Triangle

Triangle HKL (or HIL) has a 30° angle at H, right angle at I (or L), and the side opposite 30° is 22 cm (since JK is 22 cm, and IL is equal to JK, as they are both vertical segments).

Step2: Apply 30-60-90 Triangle Ratio

In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse. Let \( HK \) be the hypotenuse, and the side opposite 30° (angle at H) is 22 cm. So:

$$ \sin(30°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{22}{HK} $$

Since \( \sin(30°) = \frac{1}{2} \), we have:

$$ \frac{1}{2} = \frac{22}{HK} $$

Solving for \( HK \):

$$ HK = 22 \times 2 = 44 $$

Answer:

The length of segment HK is \(\boxed{44}\) cm.