In a right-angled triangle, the longest side has a length of 300 mm, while the horizontal length is 250 mm. Calculate the vertical length and both included angles.

Here are the solutions to your questions: 2. Calculate the vertical length and both included angles. Step 1: Use the Pythagorean theorem to find the vertical length. Let the longest side (hypotenuse) be c = 300 mm and the horizontal length be a = 250 mm. Let the vertical length be b. The Pythagorean theorem states a^2 + b^2 = c^2. (250\, mm)^2 + b^2 = (300\, mm)^2 62500\, mm^2 + b^2 = 90000\, mm^2 b^2 = 90000\, mm^2 - 62500\, mm^2 b^2 = 27500\, mm^2 b = sqrt(27500)\, mm b ≈ 165.83\, mm The vertical length is approximately 165.83 mm. Step 2: Calculate the included angles using trigonometric ratios. Let _1 be the angle opposite the vertical side (b) and adjacent to the horizontal side (a). (_1) = adjacenthypotenuse = 250\, mm300\, mm (_1) = (25)/(30) = (5)/(6) _1 = ((5)/(6)) _1 ≈ 33.56^ Let _2 be the angle opposite the horizontal side (a) and adjacent to the vertical side (b). (_2) = oppositehypotenuse = 250\, mm300\, mm (_2) = (5)/(6) _2 = ((5)/(6)) _2 ≈ 56.44^ Alternatively, since the sum of angles in a triangle is 180^ and one angle is 90^: _2 = 180^ - 90^ - _1 = 90^ - 33.56^ ≈ 56.44^ The vertical length is approximately 165.83 mm, and the included angles are approximately 33.56^ and 56.44^. 3. Work out the longest side of a right-angled triangle if the shorter sides are 50 mm and 220 mm. Step 1: Use the Pythagorean theorem. Let the shorter sides be a = 50 mm and b = 220 mm. Let the longest side (hypotenuse) be c. c^2 = a^2 + b^2 c^2 = (50\, mm)^2 + (220\, mm)^2 c^2 = 2500\, mm^2 + 48400\, mm^2 c^2 = 50900\, mm^2 c = sqrt(50900)\, mm c ≈ 225.61\, mm The longest side is approximately 225.61\, mm. 4. Work out the shorter side of a right-angled triangle if the longest side is 120 mm and the other shorter side is 90 mm. Step 1: Use the Pythagorean theorem. Let the longest side (hypotenuse) be c = 120 mm and one shorter side be a = 90 mm. Let the other shorter side be b. a^2 + b^2 = c^2 (90\, mm)^2 + b^2 = (120\, mm)^2 8100\, mm^2 + b^2 = 14400\, mm^2 b^2 = 14400\, mm^2 - 8100\, mm^2 b^2 = 6300\, mm^2 b = sqrt(6300)\, mm b ≈ 79.37\, mm The shorter side is approximately 79.37\, mm. 5. Use your calculator to work out the ratios for these angles: 5.1 sin 20° (20^) ≈ 0.3420 5.2 cos 30° (30^) ≈ 0.8660 5.3 tan 45° (45^) = 1 6. Use your calculator to work out the angles for these ratios: 6.1 sin x = 0.8337 x = (0.8337) x ≈ 56.48^ 6.2 cos x = 8/14 x = (8)/(14) = (4)/(7) x = ((4)/(7)) x ≈ 55.15^ 6.3 tan x = 32 x = (32) x ≈ 88.21^ Drop the next question! 📸