Question

1 angle b is an acute angle in a right triangle. what is a reasonable approximation of the measure of angle b if the ratio for the opposite leg to the hypotenuse is 0.67?
2 here is a triangle abc.
image of right triangle abc with right angle at b
estimate the values to complete the table.

angle	adjacent leg ÷ hypotenuse	opposite leg ÷ hypotenuse	opposite leg ÷ adjacent leg
c	0.97	0.26	0.27

3 priya says, “i know everything about a right triangle with a 30° angle and a hypotenuse with length 1 cm. here, look.”

• the other angle is 60°.
• the leg adjacent to the 30° angle is 0.866 cm long.
• the side opposite the 30° angle is 0.5 cm long.

han asks, “what would happen if a right triangle with a 30° angle has a hypotenuse that is 2 cm long instead?”
help them find the unknown angle measures and side lengths in the new triangle.
explain or show your reasoning.

Explanation:

Step1: Identify sine of angle B

The ratio of opposite leg to hypotenuse is $\sin(B) = 0.67$.

Step2: Calculate angle B

Use inverse sine: $B = \arcsin(0.67)$
$B \approx 42^\circ$

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Step1: Relate angle A to angle C

In right triangle $ABC$, $\angle A + \angle C = 90^\circ$.

Step2: Find ratios for angle A

• Adjacent/Hypotenuse for $\angle A$ = Opposite/Hypotenuse for $\angle C$ = 0.26
• Opposite/Hypotenuse for $\angle A$ = Adjacent/Hypotenuse for $\angle C$ = 0.97
• Opposite/Adjacent for $\angle A$ = $\frac{0.97}{0.26} \approx 3.73$

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Step1: Determine unknown angle

Triangle angles sum to $180^\circ$, so the other acute angle remains $60^\circ$.

Step2: Scale side lengths

The triangle is similar (scale factor = $\frac{2}{1}=2$). Multiply original sides by 2:

• Leg adjacent to $30^\circ$: $0.866 \times 2 = 1.732$ cm
• Leg opposite to $30^\circ$: $0.5 \times 2 = 1$ cm
• Hypotenuse = 2 cm (given)

Answer:

1. $\boldsymbol{42^\circ}$ (approximate)

2.

Angle	Adjacent Leg $\div$ Hypotenuse	Opposite Leg $\div$ Hypotenuse	Opposite Leg $\div$ Adjacent leg
$C$	0.97	0.26	0.27

3.

• Unknown angle: $\boldsymbol{60^\circ}$
• Leg adjacent to $30^\circ$: $\boldsymbol{1.732}$ cm
• Leg opposite to $30^\circ$: $\boldsymbol{1}$ cm
• Hypotenuse: $\boldsymbol{2}$ cm

Reasoning: The triangles are similar, so angles stay the same, and all side lengths scale by the factor of 2 (the ratio of the new hypotenuse to the original hypotenuse).