Question

find the lengths of r, s, t, and u shown in the figure below if r + s = 33. round your answers to the nearest tenth. note that the figure is not drawn to scale.
(figure with a right triangle on top with base 29, height s, hypotenuse u; and a right triangle below with hypotenuse 17, angle 43°, base t, height r; r + s = 33)
r =
s =
t =
u =

Explanation:

Step1: Analyze the smaller triangle (with hypotenuse 17)

In the smaller right triangle, we can use trigonometric ratios. Let's denote the angle as \(43^\circ\). For the side \(t\) (opposite to \(43^\circ\)) and \(r\) (adjacent to \(43^\circ\)):

• \(\sin(43^\circ)=\frac{t}{17}\), so \(t = 17\sin(43^\circ)\)
• \(\cos(43^\circ)=\frac{r}{17}\), so \(r = 17\cos(43^\circ)\)

Calculating \(r\): \(r = 17\cos(43^\circ)\approx17\times0.7314\approx12.4\) (rounded to nearest tenth)

Step2: Find \(s\) using \(r + s = 33\)

Given \(r + s = 33\), so \(s = 33 - r\). Substituting \(r\approx12.4\), we get \(s = 33 - 12.4 = 20.6\)

Step3: Analyze the larger triangle (with base 29)

For the larger right triangle, we can find \(u\) using Pythagorean theorem or trigonometry. Also, \(t\) can be found from the smaller triangle: \(t = 17\sin(43^\circ)\approx17\times0.6820\approx11.6\) (rounded to nearest tenth)

For \(u\), using Pythagorean theorem in the larger triangle: \(u=\sqrt{29^2 + s^2}\). Substituting \(s = 20.6\), \(u=\sqrt{29^2+20.6^2}=\sqrt{841 + 424.36}=\sqrt{1265.36}\approx35.6\) (rounded to nearest tenth)

Answer:

\(r\approx12.4\), \(s = 20.6\), \(t\approx11.6\), \(u\approx35.6\)