Pythagoras - TYS 1

All the triangles in the following questions are right-angled.

Remember the hypotenuse is the side opposite the right angle
so start by writing down that pronmeral or value first and then squaring it.

Basic relationships	1. Find the value of x in the above diagram. Answer.x = 20.	2. Find the value of m in the above triangle (to 1 decimal place). Answer.m = 10.4.
	3. Find the value of g in the above diagram (correct to the nearest integer). Answer.g = 80.	4. Find the value of x in triangle ABC. Answer.x = 20.
	5. Find the value of y in the above triangle. Answer.y = 8.2.	6. Find the integer value of P in the above diagram. Answer.P = 18.0.
	7. Find the value of S in the above diagram (to 2 decimal places). Answer.S = 8.86.	8. Find the length of the side XZ in the triangle above to 2 significant figures. Answer.Side XZ = 5.3.
	9. In the triangle PQR shown above, find the length of side PR. Answer.PR = 25. Note the numbers form a triad.	10. In the triangle ABC shown above, find the length of side BC. Answer.BC = 21. Note these numbers also form a triad.
Test for a right angle.	11. Does the triangle ABC above have one angle which is a right angle? Prove your conclusion and, if it does, name the angle. Answer.Angle ACB = 90°.	12. Does the triangle DEF above have one angle which is a right angle? Prove your conclusion and, if it does, name the angle. Answer.No right angle in this triangle.
	13. Does the triangle ABC above have one angle which is a right angle? Prove your conclusion and, if it does, name the angle. Answer.No right angle in this triangle.	14. Does the triangle ABC above have one angle which is a right angle? Prove your conclusion and, if it does, name the angle. Answer.Angle JKL = 90°.
Test if a triad	15. Does the set of three numbers {14, 18, 23} form a Pythagorean triad? Answer.No: √(14² + 18²) = 22.8².	16. Is the set of numbers {6, 3, 4} a Pythagorean triad? Answer.No: 3² + 4² ≠ 6².
	17. Is it correct to describe the numbers {120, 150, 90} as forming a Pythagorean triad? Answer.Yes: 90² + 120² = 150².	18. Does the set of three numbers {64, 48, 80} form a Pythagorean triad? Answer.Yes: 48² + 64² = 80².
Applied questions	19. In the above square, the diameter is 32 cm. Find the size x of the sides to 1 decimal place. Answer.x = 22.6 cm.	20. The picture above depicts a 10 m ladder leaning against a wall. The top of the ladder is 7.20 m above the floor. How far (d) is the foot of the ladder away from the wall? (answer to the nearest cm) Answer.6.94 m.
	21. A flag pole is 8 m high. It has a 12 m long support guide attached to the top of the flag pole which is also anchored on the horizontal ground away from the base of the flag pole. (i) Draw a diagram to represent this information. (ii) How far from the base of the flagpole (to the nearest cm) is the support guide anchored into the ground? Answer.8.94 m away from the flag pole.	22. A rectangular park is surrounded by 4 streets with the intersections at A, B, C and D. AB = CD = 900 m. AD = BC = 600 m. Iggy wants to walk from B to D. How much further (to the nearest metre) does he have to walk by following the streets than by walking across the diagonal BD? Answer.Street way is 418 m longer.
	23. The triangle BCD has a right angle at C. DA = AC and BC = 12.0 cm. A point A bisects the side DC and AB = 13.0 cm. (i) Find the length of AC (to the nearest mm). (ii) Hence find the length of DB (to the nearest mm). Answer.AC = 5.0 cm. DB = 15.6 cm.	24. Find the altitude (i.e. height) of an equilateral triangle having side lengths of 15 cm (correct to the nearest mm). Answer.Height is 13.0 cm.
	25. A rhombus has sides of 20 cm and the shortest diagonal measures 15 cm. How long (to the nearest cm) is the longer diagonal? Answer.Longest diagonal is 37 cm.	26. A cone has a vertical height of 15 cm while its slant height is 17 cm. How long is the diameter of the base? Answer.Diameter is 16 cm.
	27. Triangle ABC is an acute angled triangle. AD is drawn perpendicular to BC. Prove that AB² + DC² = AC² + DB² Hint:Write down two pythagorean relationships and eliminate the common squares.	28. A rectangular prism is shown in the above diagram. AB = 5 cm, BC = 10 cm and AE = 7 cm. Find the lengths (to the nearest mm) of: (i) diagonal AC. (ii) diagonal EC. Answer.(i) AC = 11.2 cm. (ii) EC = 13.2 cm.
	29. A Starship SN 10 trial launch was held on 4 March 2021. After 3 minutes 10 seconds, the starship was at an altitude of 8 km. The observation camera was 4.6 km from the centre of the launch pad. When Starship was at 8 km: (i) what had been its average speed during the flight to that time (nearest m/sec)? (ii) what was the distance between Starship and the observation camera (nearest 100 m)? Suggestion.Move the time bar along to nearly 0.00 to see the launch then along to nearly +3:00 to see the 8 km reached. Answer.(i) Average speed: 42 m/sec. (ii) distance = 9.2 km.	30. Triangle XYZ is drawn with a right angle at X. The point M is located somewhere on side XY and N is located somewhere on XZ. Prove that: NY² + MZ² = YZ² + MN² Hint:Write down the four pythagorean relationships and eliminate the square terms not included in the final relationship.
Old classics. Q31: Taken from the Chinese manuscript "Arithmetic in Nine Sections" written in the Han period (206 BC - 222 AD).	31. A circular well has a diameter of 10 m. A rigid papyrus reed is growing in the centre of the well and it projects out of the water by 1 m. When the reed is bent over, it just touches the ede of the pond. How deep is the water in the well? Answer.The water is 12 m deep.
Q32: Taken from a question posed about 630 AD by the Hindu mathematician Brahmagupta.	32. A bamboo pole 18 m high is broken by a strong wind. The top section is still joined but it lies down and the top just touches the ground 6 m from the base. What are the lengths of the two segments of the bamboo? Answer.Top section is 10 m while the lower section is 8 m.