From a point located at a distance of 12 cm from a straight line, two oblique ones are drawn to it
From a point located at a distance of 12 cm from a straight line, two oblique ones are drawn to it, forming from a right angle 45 ° and 60 °. Find the lengths of the oblique and their projections on the line.
Let from point B, located at a distance of BA = 12 cm from the straight line AK, two oblique BC and BK are drawn to it, which form angles ∠BCA = 45 ° and ∠BKA = 60 ° with the right AK. Consider a right-angled triangle ABC, in it ∠CBA = ∠BCA = 45 °, since the sum of acute angles is 90 °, which means that it is isosceles, then CA = AB = 12 cm.The hypotenuse CB can be found using the Pythagorean theorem:
CA² + BA² = СB²;
12² + 12² = СB²;
СB² = 288;
СB ≈ 16.97 (cm).
Consider a right-angled triangle ABK, in it ∠КВА = 90 ° – ∠ВКА = 90 ° – 60 ° = 30 °, since the sum of acute angles is 90 °, which means that it is isosceles, then
KA = BA ∙ tg30 °;
KA = 12 ∙ 0.5774;
KA ≈ 6.92 (cm); we get
BK = 2 ∙ KA; BK ≈ 2 ∙ 6.92; BK ≈ 13.84 (cm).
Answer: the lengths of the oblique СB ≈ 16.97 cm and BК ≈ 13.84 cm and the lengths of their projections onto the straight line CA = 12 cm and the spacecraft ≈ 6.92 cm.
