Question

mvusd, math ii
re-engagement,

1. a) find the coordinates of each vertex of the triangle. $d_{\frac{1}{3}}(\triangle abc)$, given $a(6, -2)$, $b(-9, -1)$, $c(0, -3)$.

(with a coordinate grid image)
b) what is true of the corresponding side lengths for these triangles?
c) what is true of the corresponding angles?

1. what is the length of the lake in each diagram?

(with a diagram of two triangles with side lengths 2 mi, 3 mi, 3.5 mi, 10.5 mi, 9 mi)
______ mi

1. show the ratios in the diagrams for the special right triangles below.

(with diagrams of a 30-60-90 triangle and a 45-45-90 triangle)
29-32) solve for the indicated variables. write answers as exact and approximate decimal values when appropriate.

1. (with a diagram of a 45-45-90 triangle with legs 7m, 7m, hypotenuse x)
2. (with a diagram of a 30-60-90 triangle with hypotenuse 20, side g, side f)
3. (with a diagram of a 45-45-90 triangle with hypotenuse 7, legs x, x)
4. (with a diagram of a 30-60-90 triangle with leg 12 cm, side x, side y)

Explanation:

Question 27:

Step1: Identify Similar Triangles

The two triangles are similar (vertical angles are equal, so by AA similarity). Let the length of the lake be \( x \). The sides of the smaller triangle are 2 mi, 3 mi, 3.5 mi. The sides of the larger triangle (excluding the lake) are 10.5 mi, 9 mi, and \( x \). We can set up proportions using corresponding sides. Let's use the sides 3 mi (smaller) and 10.5 mi (larger), and 2 mi (smaller) and \( x \) (larger), or 3.5 mi (smaller) and 9 mi (larger). Let's check the ratio of 3 and 10.5: \( \frac{10.5}{3} = 3.5 \). Now check 3.5 and 9: \( \frac{9}{3.5} \approx 2.57 \), no. Wait, maybe 3 and 9? \( \frac{9}{3}=3 \), 3.5 and 10.5: \( \frac{10.5}{3.5}=3 \). Ah, correct. So the ratio of larger to smaller is 3. So the side corresponding to 2 mi (smaller) is \( x \) (larger). So \( x = 2 \times 3 = 6 \)? Wait, no. Wait, the smaller triangle has sides 2, 3, 3.5. The larger has 10.5, 9, \( x \). Let's find the ratio between corresponding sides. Let's take 3 (smaller) and 9 (larger): \( \frac{9}{3} = 3 \). 3.5 (smaller) and 10.5 (larger): \( \frac{10.5}{3.5} = 3 \). So the scale factor is 3. Therefore, the side corresponding to 2 mi (smaller) is \( x \) (larger), so \( x = 2 \times 3 = 6 \)? Wait, no, wait the lake is the base of the larger triangle? Wait, the diagram: the two triangles are intersecting, forming a figure like a bowtie. The smaller triangle has top side 2, left 3, right 3.5. The larger triangle has left 10.5, right 9, and bottom (lake) \( x \). So the corresponding sides: left side of smaller (3) corresponds to left side of larger (10.5): ratio \( 10.5 / 3 = 3.5 \). Right side of smaller (3.5) corresponds to right side of larger (9): \( 9 / 3.5 \approx 2.57 \). Wait, maybe I got the correspondence wrong. Wait, vertical angles, so the angle between 3 and 3.5 in the smaller triangle is equal to the angle between 10.5 and 9 in the larger triangle. So the sides adjacent to the vertical angle: 3 and 10.5, 3.5 and 9, and 2 and \( x \). Let's check the ratio of 3 and 10.5: \( 10.5 = 3 \times 3.5 \). 3.5 and 9: \( 9 = 3.5 \times \frac{18}{7} \approx 2.57 \). No, maybe the correct correspondence is 3 (smaller) with 9 (larger), 3.5 (smaller) with 10.5 (larger), and 2 (smaller) with \( x \) (larger). Let's check \( 10.5 / 3.5 = 3 \), \( 9 / 3 = 3 \). Ah, there we go. So 3.5 (smaller right) corresponds to 10.5 (larger left): \( 10.5 / 3.5 = 3 \). 3 (smaller left) corresponds to 9 (larger right): \( 9 / 3 = 3 \). So the scale factor is 3. Therefore, the top side of the smaller triangle (2 mi) corresponds to the bottom side of the larger triangle (the lake, \( x \)): \( x = 2 \times 3 = 6 \)? Wait, no, wait the smaller triangle's top side is 2, and the larger triangle's bottom side is \( x \). Since the scale factor is 3 (larger to smaller: 10.5 / 3.5 = 3, 9 / 3 = 3), then \( x = 2 \times 3 = 6 \)? Wait, but let's check with the other ratio. Wait, maybe the smaller triangle has sides 2, 3, 3.5, and the larger has sides \( x \), 9, 10.5. So the ratio of 3 to 9 is 1/3, 3.5 to 10.5 is 1/3, so the ratio of smaller to larger is 1/3. Therefore, 2 (smaller) to \( x \) (larger) is 1/3, so \( x = 2 \times 3 = 6 \)? Wait, no, if smaller to larger is 1/3, then larger is 3 times smaller. So 2 (smaller) times 3 is 6? Wait, but let's do it properly. Let the smaller triangle have sides \( a = 2 \), \( b = 3 \), \( c = 3.5 \). The larger triangle has sides \( A = x \), \( B = 9 \), \( C = 10.5 \). Since they are similar, \( \frac{A}{a} = \frac{B}{b} = \frac{C}{c} \). Let's check \( \frac{B}{b} = \frac{9}{3} = 3 \), \( \frac…

Step1: Identify Triangle Type

This is a 45-45-90 triangle (right triangle with two 45° angles), so the legs are equal, and the hypotenuse \( x \) is \( \text{leg} \times \sqrt{2} \). The legs are both 7 m.

Step2: Calculate Hypotenuse

Using the formula for 45-45-90 triangle: \( x = 7\sqrt{2} \approx 7 \times 1.414 \approx 9.899 \) m.

Step1: Identify Triangle Type

This is a 30-60-90 triangle? Wait, no, it's a right triangle with angle 60°, so the other angle is 30°. 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. Wait, here the hypotenuse? Wait, the triangle has a right angle, angle 60°, and side 20 (opposite 30°? Wait, no. Let's label the triangle: right angle at the top, angle 60° at the bottom, side 20 is the hypotenuse? Wait, no, the side labeled 20 is opposite the right angle? No, the right angle is at the top, so the sides: \( g \) is the hypotenuse? Wait, no, the right angle is between \( f \) and the side adjacent to 60°. Wait, let's denote: right angle at \( C \), angle at \( B \) is 60°, side \( AB = 20 \) (hypotenuse), side \( BC = f \) (adjacent to 60°), side \( AC = g \) (opposite to 60°). Wait, no, the triangle is labeled with side 20, angle 60°, right angle. So cos(60°) = adjacent / hypotenuse, sin(60°) = opposite / hypotenuse. Wait, if the side opposite 60° is \( f \), and adjacent is \( g \)? No, let's see: the side labeled 20 is the hypotenuse? Wait, no, in the diagram, the side labeled 20 is one of the legs? Wait, the triangle has a right angle, angle 60°, and side 20 (let's say it's the hypotenuse). Then:

• \( \cos(60°) = \frac{\text{adjacent}}{20} \), so adjacent (let's say \( f \)) = \( 20 \times \cos(60°) = 20 \times 0.5 = 10 \).
• \( \sin(60°) = \frac{\text{opposite}}{20} \), so opposite (let's say \( g \)) = \( 20 \times \sin(60°) = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.32 \).

Wait, but maybe the side 20 is opposite the 30° angle. Wait, no, angle 60°, so the other angle is 30°. In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse. So if side 20 is opposite 30°, then hypotenuse \( g = 40 \), and side \( f \) (opposite 60°) is \( 20\sqrt{3} \approx 34.64 \). Wait, this is confusing. Wait, the diagram: right angle at the top, angle 60° at the bottom, side 20 is the side opposite the 60° angle? No, let's look at the labels: the side labeled 20 is the one with angle 60° adjacent to it. Wait, maybe the triangle has: right angle, angle 60°, side 20 (adjacent to 60°), side \( f \) (opposite to 60°), and hypotenuse \( g \). Then:

• \( \cos(60°) = \frac{20}{g} \implies g = \frac{20}{\cos(60°)} = \frac{20}{0.5} = 40 \)
• \( \sin(60°) = \frac{f}{g} \implies f = 40 \times \sin(60°) = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \approx 34.64 \)

Yes, that makes sense. So \( f = 10 \) (wait, no, earlier mistake). Wait, no, if the side adjacent to 60° is 20, then:

\( \cos(60°) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{20}{g} \implies g = 20 / 0.5 = 40 \)

\( \sin(60°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{f}{g} \implies f = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \approx 34.64 \)

Answer:

6

Question 29: