Question

for the right triangles below, find the exact values of the side lengths c and d. if necessary, write your responses in simplified radical form. there are two right triangle diagrams: one with angles 60° and 30°, leg length 2, and side c; another with angles 45° and 45°, hypotenuse length 3, and leg d. there are input boxes for c and d, and a keypad with radical and fraction buttons.

Explanation:

Step1: Find \( c \) (30-60-90 triangle)

In a 30-60-90 triangle, the sides are in the ratio \( 1 : \sqrt{3} : 2 \), where the side opposite 30° is the shortest, opposite 60° is \( \sqrt{3} \) times that, and hypotenuse is twice the shortest. Here, the side opposite 60° is 2? Wait, no—wait, the right angle, 60°, 30°. The side with length 2 is adjacent to 60°? Wait, no, let's label: the right angle, one angle 60°, one 30°. The side of length 2 is adjacent to 60° (so it's the side opposite 30°? Wait, no: in a right triangle, angles sum to 180, so right angle (90), 60, 30. So the side opposite 30° is the shortest. Wait, the side labeled 2: let's see, the angle at the top is 60°, right angle at the bottom left, so the side of length 2 is adjacent to 60°, so it's the side opposite 30°? Wait, no, opposite 30° would be the side opposite the 30° angle. The 30° angle is at the bottom right. So the side opposite 30° is the side of length 2? Wait, no, the side adjacent to 30° is the longer leg? Wait, no, in 30-60-90: hypotenuse = 2 * shorter leg (opposite 30°), longer leg (opposite 60°) = shorter leg * \( \sqrt{3} \). So if the side opposite 30° is \( x \), then hypotenuse is \( 2x \), longer leg is \( x\sqrt{3} \). Wait, in the first triangle, the side with length 2: let's see, the angle at the top is 60°, so the side opposite 60° is \( c \)? No, wait, the right angle is at the bottom left, so the sides: vertical side is 2, horizontal side is \( c \), hypotenuse is... Wait, no, the angles: 60° at the top, 30° at the bottom right. So the side opposite 30° is the vertical side (length 2), so hypotenuse is \( 2*2 = 4 \)? No, wait, no—wait, the vertical side is length 2, adjacent to 30°? Wait, I'm confused. Wait, let's use trigonometry. \( \tan(60°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{c}{2} \). \( \tan(60°) = \sqrt{3} \), so \( c = 2\sqrt{3} \)? Wait, no, \( \tan(60) = \sqrt{3} = \frac{c}{2} \), so \( c = 2\sqrt{3} \)? Wait, no, wait: angle 60° is at the top, so the opposite side is \( c \), adjacent is 2. So \( \tan(60) = \frac{c}{2} \), so \( c = 2\tan(60) = 2\sqrt{3} \). Wait, but maybe it's a 30-60-90 triangle where the shorter leg (opposite 30°) is 2? Wait, no, if the shorter leg (opposite 30°) is 2, then hypotenuse is 4, longer leg (opposite 60°) is \( 2\sqrt{3} \). Wait, maybe I mixed up. Let's do the second triangle first.

Step2: Find \( d \) (45-45-90 triangle)

In a 45-45-90 triangle, the legs are equal, and hypotenuse is leg * \( \sqrt{2} \). Here, hypotenuse is 3, so leg \( d = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \)? Wait, no—wait, the triangle has two 45° angles, so it's isoceles right triangle. So legs are equal, hypotenuse \( h = d\sqrt{2} \). So \( d = \frac{h}{\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \). Wait, but let's check: if legs are \( d \), hypotenuse is \( d\sqrt{2} = 3 \), so \( d = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \).

Wait, back to first triangle: 30-60-90. Let's re-express. Angles: 90, 60, 30. So sides: let’s say the side opposite 30° is \( x \), then hypotenuse is \( 2x \), side opposite 60° is \( x\sqrt{3} \). In the first triangle, the side with length 2: is it the side opposite 60°? Wait, the angle of 60°: the side opposite 60° would be \( x\sqrt{3} \), and side opposite 30° is \( x \), hypotenuse \( 2x \). Wait, maybe the side of length 2 is the side opposite 30°? Then hypotenuse is \( 4 \), and side opposite 60° is \( 2\sqrt{3} \). But in the diagram, the side labeled \( c \) is adjacent to 30°? Wait, no, the first triangle: right angle at bottom left, 60° at top, 30° at bottom right. So the side from bottom left to bottom right is \( c \) (horizontal), bottom left to top is 2 (vertical), top to bottom right is hypotenuse. So angle at bottom right is 30°, so the side opposite 30° is the vertical side (length 2), so hypotenuse is \( 2*2 = 4 \), and the side \( c \) (opposite 60°) is \( 2\sqrt{3} \). So \( c = 2\sqrt{3} \).

Wait, let's confirm with trigonometry: \( \cos(30°) = \frac{\text{adjacent}}{\text{hypotenuse}} \), but maybe \( \tan(30°) = \frac{2}{c} \), since \( \tan(30°) = \frac{1}{\sqrt{3}} = \frac{2}{c} \), so \( c = 2\sqrt{3} \). Yes, that makes sense.

For \( d \): 45-45-90 triangle, hypotenuse 3, so leg \( d = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \) (rationalizing the denominator).

Answer:

\( c = 2\sqrt{3} \)
\( d = \frac{3\sqrt{2}}{2} \)