Question

1. express the pattern in terms of the x given on the triangle.
2. describe the pattern.

(images of triangles: one right triangle with 30°, one triangle with sides 7√2, 14√2, and a sketchy triangle)

Explanation:

Problem 19: Express the pattern in terms of \( x \) given on the triangle (assuming it's a 30-60-90 right triangle)

Step 1: Recall 30-60-90 triangle ratios

In a 30-60-90 right triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \), where:

• The side opposite \( 30^\circ \) (shorter leg) is \( s \),
• The side opposite \( 60^\circ \) (longer leg) is \( s\sqrt{3} \),
• The hypotenuse is \( 2s \).

In the given triangle, the angle is \( 30^\circ \), and \( x \) is the side opposite \( 30^\circ \) (shorter leg). Let’s denote the hypotenuse as \( h \), longer leg as \( l \).

Step 2: Relate sides using the ratio

If the shorter leg (opposite \( 30^\circ \)) is \( x \), then:

• Hypotenuse \( h = 2x \),
• Longer leg \( l = x\sqrt{3} \).

Assuming the other triangle (with sides \( 7\sqrt{2} \), \( 14\sqrt{2} \)) is a similar 30-60-90 triangle (checking ratios: \( \frac{14\sqrt{2}}{7\sqrt{2}} = 2 \), so hypotenuse is twice the shorter leg). Thus, the pattern follows the 30-60-90 triangle side ratios.

Problem 20: Describe the pattern

The triangles follow the 30-60-90 right triangle side ratio pattern (\( 1 : \sqrt{3} : 2 \)). For the triangle with sides \( 7\sqrt{2} \), \( 14\sqrt{2} \), the hypotenuse (\( 14\sqrt{2} \)) is twice the shorter leg (\( 7\sqrt{2} \)), matching the 30-60-90 ratio. The first triangle (with \( x \)) also follows this: shorter leg \( x \), hypotenuse \( 2x \), longer leg \( x\sqrt{3} \). The pattern is consistent with 30-60-90 triangle side length relationships (shorter leg \( s \), longer leg \( s\sqrt{3} \), hypotenuse \( 2s \)).

Answer:

s:

1. In a 30-60-90 triangle, if \( x \) is the shorter leg (opposite \( 30^\circ \)):

• Hypotenuse \( = 2x \),
• Longer leg \( = x\sqrt{3} \).

(Pattern: Sides are \( x \), \( x\sqrt{3} \), \( 2x \) for 30-60-90 triangle.)

1. The pattern is that the triangles are 30-60-90 right triangles, where the hypotenuse is twice the shorter leg, and the longer leg is \( \sqrt{3} \) times the shorter leg (consistent with \( 1 : \sqrt{3} : 2 \) side ratio).