Question

3.13.2 tangled triangles
determine the scale factor for each pair of triangles.
(hint: ef = 16. use this and the pythagorean theorem to find hf)

	ratio	correct?
δb to δc	0.8	yes
δa to δc		try again...

reset

(images of three right triangles: triangle a with hypotenuse 12, angle 53°; triangle b with hypotenuse 18, angle 37°; triangle c with base 12, angle 53°, angle 37° at f, ef=16)

Explanation:

Step1: Recall Scale Factor Definition

Scale factor from \(\Delta A\) to \(\Delta C\) is \(\frac{\text{Side of } \Delta C}{\text{Side of } \Delta A}\). First, find corresponding sides. From \(\Delta B\) to \(\Delta A\), scale factor is \(0.75=\frac{3}{4}\), so if \(\Delta B\) side is 18, \(\Delta A\) side (hypotenuse) is 12 (given). From \(\Delta B\) to \(\Delta C\), scale factor is \(0.8 = \frac{4}{5}\), so \(\Delta C\) hypotenuse: \(18\div0.8 = 22.5\) (or using \(\Delta A\) hypotenuse 12, \(\Delta C\) hypotenuse: \(12\div0.75\div0.8=20?\) Wait, better: find a side. Let's use EF = 16 (hint). Wait, in \(\Delta B\), let's find EG (leg). \(\Delta B\) has hypotenuse 18, angle 37°, so EG (opposite 37°? Wait, no, right triangle: angles 37°, 53°, 90°. So in \(\Delta A\), hypotenuse 12, \(\Delta B\) hypotenuse 18, scale factor \(\Delta B\) to \(\Delta A\) is \(12/18 = 2/3\)? Wait no, earlier table says \(\Delta B\) to \(\Delta A\) is 0.75? Wait maybe I mixed. Wait the table says \(\Delta B\) to \(\Delta A\) ratio 0.75, so \(\text{Side of } A = 0.75 \times \text{Side of } B\). So if \(\Delta B\) hypotenuse is 18, then \(\Delta A\) hypotenuse is \(18 \times 0.75 = 13.5\)? Wait no, the diagram shows \(\Delta A\) hypotenuse 12, \(\Delta B\) hypotenuse 18. So 12/18 = 2/3 ≈ 0.666, but table says 0.75. Wait maybe the sides: in \(\Delta A\), hypotenuse 12, \(\Delta B\) hypotenuse 18. Wait the hint says EF = 16. Wait EF is a leg? Let's look at \(\Delta C\): EH = 12 (given). EF = 16 (hint). So in \(\Delta C\), right triangle with leg EH = 12, leg EF = 16, so hypotenuse HF: by Pythagoras, \(HF = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20\). Ah! So \(\Delta C\) hypotenuse HF = 20. Now, \(\Delta A\) hypotenuse: let's see \(\Delta B\) to \(\Delta A\) ratio 0.75, \(\Delta B\) hypotenuse 18, so \(\Delta A\) hypotenuse: \(18 \times 0.75 = 13.5\)? No, but \(\Delta C\) hypotenuse is 20. Wait no, in \(\Delta A\), the hypotenuse is FG? Wait the diagram: \(\Delta A\) has vertices F, G, H: right angle at G, angle 53° at H, so hypotenuse FH = 12. \(\Delta B\): F, G, E: right angle at G, hypotenuse FE = 18. \(\Delta C\): F, E, H: right angle at E, leg EH = 12, leg EF = 16, hypotenuse FH = 20 (since 12-16-20 triangle, 3-4-5 scaled by 4). Ah! So \(\Delta C\) hypotenuse FH = 20. Now, \(\Delta A\) hypotenuse FH (wait no, \(\Delta A\) is F, G, H: hypotenuse FH = 12? Wait no, in \(\Delta A\), F, G, H: right angle at G, so FH is hypotenuse, length 12. \(\Delta C\) is F, E, H: right angle at E, hypotenuse FH = 20. So scale factor \(\Delta A\) to \(\Delta C\) is \(\text{Side of } C / \text{Side of } A = 20 / 12 = 5/3 ≈ 1.666\)? Wait no, scale factor from A to C is (side of C)/(side of A). Wait \(\Delta A\) hypotenuse 12, \(\Delta C\) hypotenuse 20. So 20/12 = 5/3 ≈ 1.666, or 12/20 = 3/5 = 0.6? Wait no, let's check the table. \(\Delta B\) to \(\Delta A\) is 0.75 (so A = 0.75 B), \(\Delta B\) to \(\Delta C\) is 0.8 (so C = B / 0.8 = 1.25 B). Then A to C: (1.25 B) / (0.75 B) = 1.25 / 0.75 = 5/3 ≈ 1.666, or 0.75 / 0.8 = 0.9375? No, wait: if \(\Delta B\) to \(\Delta A\) is 0.75 (A = 0.75 B), \(\Delta B\) to \(\Delta C\) is 0.8 (C = B / 0.8 = 1.25 B). So A to C: C / A = (1.25 B) / (0.75 B) = 1.25 / 0.75 = 5/3 ≈ 1.666, or 4/3 * 5/4 = 5/3? Wait using the hypotenuse: \(\Delta A\) hypotenuse 12, \(\Delta C\) hypotenuse 20. 20/12 = 5/3 ≈ 1.666. Alternatively, using the leg: in \(\Delta C\), EH = 12, in \(\Delta A\), GH: let's find GH. \(\Delta A\) hypotenuse 12, angle 53° at H, so GH (adjacent to 53°) is \(12 \times \cos(53°) ≈ 12 \times 0.6 = 7.2\). \(\Delta C\) has EH = 12 (adjacent to 53°? Wait \(\Delta C\) angle at H is 53°, so EH is adjacent, FH is hypotenuse 20, so EH = 20 \times \cos(53°) ≈ 20 \times 0.6 = 12, which matches. So GH in \(\Delta A\) is 7.2, EH in \(\Delta C\) is 12. So scale factor A to C: 12 / 7.2 = 5/3 ≈ 1.666. So 5/3 or 1.666...

Step2: Calculate Scale Factor

Scale factor from \(\Delta A\) to \(\Delta C\) is \(\frac{\text{Length of corresponding side in } \Delta C}{\text{Length of corresponding side in } \Delta A}\). Using hypotenuses: \(\Delta A\) hypotenuse = 12, \(\Delta C\) hypotenuse = 20 (from 12-16-20 triangle, since EF=16, EH=12, so HF=\(\sqrt{12^2 + 16^2}\)=20). Thus, scale factor = \(20 / 12 = \frac{5}{3} \approx 1.666...\) or as a fraction \(\frac{5}{3}\) or decimal ≈ 1.6667. Alternatively, using the scale factors from the table: \(\Delta B\) to \(\Delta A\) is 0.75, \(\Delta B\) to \(\Delta C\) is 0.8, so \(\Delta A\) to \(\Delta C\) is \(0.8 / 0.75 = \frac{8}{10} / \frac{3}{4} = \frac{4}{5} \times \frac{4}{3} = \frac{16}{15}\)? Wait no, that's wrong. Wait if \(\Delta B\) to \(\Delta A\) is 0.75 (A = 0.75 B), and \(\Delta B\) to \(\Delta C\) is 0.8 (C = B / 0.8 = 1.25 B), then A to C is C / A = (1.25 B) / (0.75 B) = 1.25 / 0.75 = 5/3 ≈ 1.666..., which matches the hypotenuse method. So the correct scale factor is \(\frac{5}{3}\) or approximately 1.6667.

Answer:

\(\frac{5}{3}\) (or approximately \(1.67\))