The hypotenuse and Pythagoras' theorem
To check whether a room corner is truly a right angle, a carpenter marks 60 cm along one wall and 80 cm along the other, then looks at the slanting distance between the two marks. If it comes to 100 cm, the corner is a right angle. This is a special relationship between the sides of a right-angled triangle.
The longest side of a right-angled triangle, the one facing the right angle, is named the hypotenuse. Pythagoras' theorem: (hypotenuse)² = (one side)² + (other side)². Here the two sides are the ones meeting at the right angle.
In the figure: The same triangle is placed in three ways. The sides 6 and 8 form the right angle. However the triangle is turned, the side opposite the right angle, 10, is the hypotenuse.
Rule 1: The hypotenuse is the side opposite the right angle and the longest side — Look at which corner has the right angle. The side facing it is the hypotenuse. The hypotenuse is always longer than both other sides.
- Easy: The sides of a right-angled triangle are 6 cm, 8 cm and 10 cm. The longest side is 10, so the hypotenuse is 10 cm.
- Medium: The three sides of a right-angled triangle are 20, 21 and 29. The longest is 29, so the hypotenuse is 29.
- Exam type: The hypotenuse is 50 cm and one side is 14 cm. The other side must be shorter than 50, so it cannot be 52.
In the figure: The sides forming the right angle are 6 and 8; the hypotenuse is 10. The square on side 6 has 36 small squares, the square on side 8 has 64, and the square on the hypotenuse has area 100. So 36 + 64 = 100.
Rule 2: The hypotenuse squared equals the other two sides squared and added — Square each of the two sides on its own, then add. The total equals the hypotenuse squared.
- Easy: Sides 6 and 8, hypotenuse 10. 6² + 8² = 36 + 64 = 100 = 10².
- Medium: Sides 15 and 20, hypotenuse 25. 15² + 20² = 225 + 400 = 625 = 25².
- Exam type: Sides 0.6 and 0.8, hypotenuse 1. 0.6² + 0.8² = 0.36 + 0.64 = 1 = 1².
In the figure: 4 equal triangles with sides 6, 8, 10 are placed in a square of side 14 in two ways. Each triangle has area 24. In the first arrangement the middle square is 100, so 100 + 4 × 24 = 196. In the second arrangement the squares are 36 and 64, so 36 + 64 + 4 × 24 = 196. Both are 196, so 100 = 36 + 64.
Rule 3: Check the theorem by measuring areas — The side of the big square is the sum of the two right-angle sides. If four triangles are taken away, the leftover area can be arranged in two ways; that is why the squares match.
- Easy: The side of the big square is 6 + 8 = 14. Its area is 14 × 14 = 196.
- Medium: The area of one triangle is ½ × 6 × 8 = 24. For four triangles, 4 × 24 = 96.
- Exam type: First arrangement 100 + 96 = 196. Second arrangement 36 + 64 + 96 = 196. So the square on the hypotenuse, 100, equals 36 + 64.
The hypotenuse and Pythagoras' theorem
In the app you can move this figure with your finger.
Open in the appThis theorem works only in a right-angled triangle. The hypotenuse is opposite the right angle and is the longest.
Adding the sides without squaring. 6 + 8 = 14, but the hypotenuse is 10. Correct: 6² + 8² = 100, so the hypotenuse is 10.
Picture the squares on the three sides: the two small squares together make the big square. In the figure, 36 + 64 = 100.
Spot the difference
| Point | Hypotenuse | Side forming the right angle |
|---|---|---|
| Position | Opposite the right angle | Touching the right angle |
| Length | The longest | Shorter than the hypotenuse |
| Square area in the figure with 6, 8, 10 | 100 | 36 and 64 |
| In the formula | (hypotenuse)² is on one side | The two squares are added |
Practice questions
- 1The sides of a right-angled triangle are 6 cm, 8 cm and 10 cm. Which is the hypotenuse?
- 2The three sides of a right-angled triangle are 30, 40 and 50. Which is the hypotenuse?
- 3Take sides 6 and 8 and hypotenuse 10. Find 6² + 8² and 10², and check the theorem.
- 4The sides forming the right angle are 18 and 24, and the hypotenuse is 30. Show that the theorem holds.
- 5The square on the hypotenuse has area 100 and the square on one side has area 36. Find the area of the square on the other side and the length of that side.
- 6Rita says, "If the sides are 6 and 8, the hypotenuse is 6 + 8 = 14." Point out her mistake and find the correct hypotenuse.
- 7The squares on two sides of a right-angled triangle have areas 225 and 400. How large is the square on the hypotenuse, and how long is the hypotenuse itself?
- 8In a square of side 14, four triangles with right-angle sides 6 and 8 (each of area 24) are placed so that the middle part is a square. What is the area of the middle square?
Show answers (try them yourself first)
- 1The longest side is 10, so the hypotenuse is 10 cm. 2) The longest side is 50, so the hypotenuse is 50. 3) 6² + 8² = 36 + 64 = 100 and 10² = 100; both are equal.
- 418² + 24² = 324 + 576 = 900 and 30² = 900; so (hypotenuse)² = the sum.
- 5100 − 36 = 64 is the area of the other square; the length of that side is 8. 6) Rita did not square the sides. Correct: 6² + 8² = 36 + 64 = 100, so the hypotenuse is 10, not 14.
- 7225 + 400 = 625 is the area of the square on the hypotenuse; the hypotenuse is 25.
- 8The big square is 14 × 14 = 196, the four triangles are 4 × 24 = 96, so the middle square is 196 − 96 = 100.
Finding the hypotenuse
To make a straight slanting path from one corner of a field to the opposite corner, you need to know its length. If the field is a rectangle, once its length and width are known, the slanting length can be found with the theorem.
To find the hypotenuse when the two sides forming the right angle are known: square both sides, add the squares, and take the square root of the sum. A square root is the number that gives the original number when multiplied by itself.
In the figure: The sides forming the right angle are 15 and 20. The square of the hypotenuse is 15² + 20² = 225 + 400 = 625. The hypotenuse is √625 = 25, because 25 × 25 = 625.
Rule 1: Square, add, take the square root — Do these three steps in order. The last step is the square root; do not forget it.
- Easy: Sides 15 cm and 20 cm. Square of the hypotenuse is 15² + 20² = 225 + 400 = 625, so the hypotenuse is √625 = 25 cm.
- Medium: Sides 20 cm and 21 cm. Square of the hypotenuse is 20² + 21² = 400 + 441 = 841, so the hypotenuse is √841 = 29 cm.
- Exam type: Sides 24 cm and 70 cm. Square of the hypotenuse is 24² + 70² = 576 + 4900 = 5476, so the hypotenuse is √5476 = 74 cm.
In the figure: Take sides 2 and 5. 2² + 5² = 4 + 25 = 29. 5² = 25 and 6² = 36; 25 < 29 < 36, so 5 < √29 < 6. The square root lies between two whole numbers.
Rule 2: If the sum is not a perfect square, the square root lies between two whole numbers — Find the perfect squares on both sides of the sum. The sum lies between them, so the square root lies between their roots.
- Easy: Sides 2 and 5. 2² + 5² = 4 + 25 = 29. 25 < 29 < 36, so 5 < √29 < 6.
- Medium: Sides 4 and 7. 4² + 7² = 16 + 49 = 65. 64 < 65 < 81, so 8 < √65 < 9.
- Exam type: Sides 3 and 6. 3² + 6² = 9 + 36 = 45. 36 < 45 < 49, so 6 < √45 < 7.
Rule 3: Bring both sides to the same unit — If one side is in centimetres and the other in metres, first change both to centimetres. Then square.
- Easy: Sides 30 cm and 0.4 metres. 0.4 metres is 40 cm. 30² + 40² = 900 + 1600 = 2500, so the hypotenuse is √2500 = 50 cm.
- Medium: Sides 80 cm and 0.6 metres. 0.6 metres is 60 cm. 80² + 60² = 6400 + 3600 = 10000, so the hypotenuse is √10000 = 100 cm, that is 1 metre.
- Exam type: Sides 1.5 metres and 200 cm. 1.5 metres is 150 cm. 150² + 200² = 22500 + 40000 = 62500, so the hypotenuse is √62500 = 250 cm, that is 2.5 metres.
Rule 4: After finding the hypotenuse, find the perimeter and area — Once the hypotenuse is known, all three sides are known. Perimeter is the sum of the three sides. Area is ½ × one right-angle side × the other right-angle side.
- Easy: Sides 18 and 24. 18² + 24² = 324 + 576 = 900, hypotenuse √900 = 30. Perimeter 18 + 24 + 30 = 72.
- Medium: Sides 16 and 30. 16² + 30² = 256 + 900 = 1156, hypotenuse √1156 = 34. Perimeter 16 + 30 + 34 = 80.
- Exam type: Sides 14 and 48. 14² + 48² = 196 + 2304 = 2500, hypotenuse √2500 = 50. Perimeter 14 + 48 + 50 = 112 and area ½ × 14 × 48 = 336.
Finding the hypotenuse
In the app you can move this figure with your finger.
Open in the appThe hypotenuse is always longer than the longer right-angle side and shorter than the sum of the two sides.
Forgetting the square root. 625 is the square of the hypotenuse; the hypotenuse is 25, not 625.
Check the answer by estimating: for sides 15 and 20, 15 + 20 = 35, so the hypotenuse must lie between 20 and 35. 25 fits.
Spot the difference
| Point | Sum is a perfect square | Sum is not a perfect square |
|---|---|---|
| Sides | 15 and 20 | 2 and 5 |
| Sum of squares | 625 | 29 |
| Square root | 25 | Between 5 and 6 |
| Are all three sides whole numbers? | Yes | No |
Practice questions
- 1The sides forming the right angle are 6 cm and 8 cm. Find the hypotenuse.
- 2The sides forming the right angle are 30 cm and 40 cm. Find the hypotenuse.
- 3The sides forming the right angle are 21 cm and 28 cm. Find the hypotenuse.
- 4The sides forming the right angle are 1.2 cm and 1.6 cm. Find the hypotenuse.
- 5The sides forming the right angle are 5 cm and 8 cm. Between which two whole numbers does the hypotenuse lie?
- 6The sides forming the right angle are 60 cm and 0.8 metres. Find the hypotenuse in cm.
- 7The sides forming the right angle are 36 cm and 48 cm. Find the hypotenuse and the perimeter.
- 8The sides forming the right angle are 20 cm and 99 cm. Find the hypotenuse and the area.
Show answers (try them yourself first)
- 16² + 8² = 36 + 64 = 100, hypotenuse √100 = 10 cm. 2) 30² + 40² = 900 + 1600 = 2500, hypotenuse √2500 = 50 cm. 3) 21² + 28² = 441 + 784 = 1225, hypotenuse √1225 = 35 cm.
- 41.2² + 1.6² = 1.44 + 2.56 = 4, hypotenuse √4 = 2 cm. 5) 5² + 8² = 25 + 64 = 89. 81 < 89 < 100, so 9 < √89 < 10. 6) 0.8 metres is 80 cm. 60² + 80² = 3600 + 6400 = 10000, hypotenuse √10000 = 100 cm.
- 736² + 48² = 1296 + 2304 = 3600, hypotenuse √3600 = 60 cm. Perimeter 36 + 48 + 60 = 144 cm.
- 820² + 99² = 400 + 9801 = 10201, hypotenuse √10201 = 101 cm. Area ½ × 20 × 99 = 990 sq cm.
Finding a side that forms the right angle
To work out how high a ladder leaning on a wall reaches, you know the ladder's length and the distance of its foot from the wall. The ladder is the hypotenuse, so the height can be found with the theorem.
To find the other side when the hypotenuse and one right-angle side are known: square the hypotenuse, subtract the square of the known side, and take the square root of the result. Here you subtract, not add.
In the figure: The hypotenuse is 50 and one side is 30. To find the other side, 50² − 30² = 2500 − 900 = 1600. The side is √1600 = 40, because 40 × 40 = 1600.
Rule 1: From the square of the hypotenuse, subtract the square of the known side — The hypotenuse is the largest, so always write its square first. Then subtract and take the square root.
- Easy: Hypotenuse 50 cm and one side 30 cm. 50² − 30² = 2500 − 900 = 1600, so the other side is √1600 = 40 cm.
- Medium: Hypotenuse 85 cm and one side 13 cm. 85² − 13² = 7225 − 169 = 7056, so the other side is √7056 = 84 cm.
- Exam type: Hypotenuse 73 cm and one side 55 cm. 73² − 55² = 5329 − 3025 = 2304, so the other side is √2304 = 48 cm.
Rule 2: Check the answer by adding back — Add the square of the side you found and the square of the known side. The sum must equal the square of the hypotenuse.
- Easy: The sides are 30 and 40. 30² + 40² = 900 + 1600 = 2500 = 50². The hypotenuse 50 came out, so the answer is right.
- Medium: Hypotenuse 97 and one side 65. 97² − 65² = 9409 − 4225 = 5184, side √5184 = 72. Check: 65² + 72² = 4225 + 5184 = 9409 = 97².
- Exam type: Hypotenuse 45 and one side 27. 45² − 27² = 2025 − 729 = 1296, side √1296 = 36. Check: 27² + 36² = 729 + 1296 = 2025 = 45².
In the figure: The hypotenuse is 9 and one side is 5. 9² − 5² = 81 − 25 = 56. 7² = 49 and 8² = 64; 49 < 56 < 64, so 7 < √56 < 8.
Rule 3: When the difference is not a perfect square — The answer lies between two whole numbers. Find the perfect squares on both sides of the difference.
- Easy: Hypotenuse 9 and side 5. 9² − 5² = 81 − 25 = 56. 49 < 56 < 64, so 7 < √56 < 8.
- Medium: Hypotenuse 12 and side 7. 12² − 7² = 144 − 49 = 95. 81 < 95 < 100, so 9 < √95 < 10.
- Exam type: Hypotenuse 20 and side 11. 20² − 11² = 400 − 121 = 279. 256 < 279 < 289, so 16 < √279 < 17.
Finding a side that forms the right angle
In the app you can move this figure with your finger.
Open in the appIf the hypotenuse is given, subtract; if both right-angle sides are given, add.
Adding the squares when the hypotenuse is given. 50² + 30² = 2500 + 900 = 3400 is wrong; the correct subtraction is 2500 − 900 = 1600.
The answer must be smaller than the hypotenuse. The side 40 is smaller than 50. Also 30 + 40 = 70 is larger than the hypotenuse 50.
Spot the difference
| Point | Finding the hypotenuse | Finding a side |
|---|---|---|
| Given | Two right-angle sides | Hypotenuse and one side |
| Operation on squares | Addition | Subtraction |
| Square in the figure | 625 | 1600 |
| Answer | Hypotenuse 25 | Side 40 |
Practice questions
- 1The hypotenuse is 100 cm and one side is 60 cm. Find the other side.
- 2The hypotenuse is 37 cm and one side is 35 cm. Find the other side.
- 3The hypotenuse is 34 cm and one side is 16 cm. Find the other side.
- 4The hypotenuse is 2.5 cm and one side is 1.5 cm. Find the other side.
- 5The hypotenuse is 15 cm and one side is 10 cm. Between which two whole numbers does the other side lie?
- 6The hypotenuse is 65 cm and one side is 33 cm. Find the other side.
- 7The hypotenuse is 130 cm and one side is 50 cm. Find the other side, the perimeter and the area.
- 8The hypotenuse is 52 cm and one side is 48 cm. Find the other side and the area.
Show answers (try them yourself first)
- 1100² − 60² = 10000 − 3600 = 6400, side √6400 = 80 cm. 2) 37² − 35² = 1369 − 1225 = 144, side √144 = 12 cm. 3) 34² − 16² = 1156 − 256 = 900, side √900 = 30 cm.
- 42.5² − 1.5² = 6.25 − 2.25 = 4, side √4 = 2 cm. 5) 15² − 10² = 225 − 100 = 125. 121 < 125 < 144, so 11 < √125 < 12. 6) 65² − 33² = 4225 − 1089 = 3136, side √3136 = 56 cm.
- 7130² − 50² = 16900 − 2500 = 14400, side √14400 = 120 cm. Perimeter 50 + 120 + 130 = 300 cm. Area ½ × 50 × 120 = 3000 sq cm.
- 852² − 48² = 2704 − 2304 = 400, side √400 = 20 cm. Area ½ × 20 × 48 = 480 sq cm.
Pythagorean triplets
Craftsmen laying out a foundation use cords of certain fixed lengths, because a triangle made from those three lengths is sure to have a right angle. Three whole numbers like these are called a Pythagorean triplet.
If the square of the largest of three natural numbers equals the sum of the squares of the other two, the three numbers are called a Pythagorean triplet. A triangle with such three sides is a right-angled triangle.
In the figure: The numbers are 15, 36 and 39. The largest is 39; its square is 39² = 1521. For the other two, 15² + 36² = 225 + 1296 = 1521. Both are equal, so this is a triplet.
Rule 1: Compare the square of the largest number with the sum of the squares of the other two — If both are equal, it is a triplet. Even a small difference means it is not.
- Easy: (28, 45, 53). 28² + 45² = 784 + 2025 = 2809 and 53² = 2809. This is a triplet.
- Medium: (39, 80, 89). 39² + 80² = 1521 + 6400 = 7921 and 89² = 7921. This is a triplet.
- Exam type: (60, 91, 109). 60² + 91² = 3600 + 8281 = 11881 and 109² = 11881. This is a triplet.
Rule 2: Even if the numbers come in any order, pick out the largest first — First decide the largest number. Keep its square on one side. Add the squares of the other two and compare.
- Easy: (52, 20, 48). The largest is 52. 20² + 48² = 400 + 2304 = 2704 and 52² = 2704. It is a triplet.
- Medium: (10, 6, 7). The largest is 10. 6² + 7² = 36 + 49 = 85, but 10² = 100. It is not a triplet.
- Exam type: (65, 33, 56). The largest is 65. 33² + 56² = 1089 + 3136 = 4225 and 65² = 4225. It is a triplet.
Rule 3: Multiplying all three numbers of a triplet by the same number gives a triplet again — Every square gets multiplied by the square of that number, so the equality stays.
- Easy: Multiply (6, 8, 10) by 3: (18, 24, 30). 18² + 24² = 324 + 576 = 900 = 30².
- Medium: Multiply (20, 21, 29) by 2: (40, 42, 58). 40² + 42² = 1600 + 1764 = 3364 = 58².
- Exam type: Multiply (15, 20, 25) by 4: (60, 80, 100). 60² + 80² = 3600 + 6400 = 10000 = 100².
In the figure: Take the number 7. 7² = 49. 49 − 1 = 48, 2 × 7 = 14 and 49 + 1 = 50. The triplet is (48, 14, 50). Check: 48² + 14² = 2304 + 196 = 2500 = 50².
Rule 4: Make a triplet from any number — Take a number greater than 1. Its square − 1, its double and its square + 1 are three numbers that form a triplet.
- Easy: Number 6. 36 − 1 = 35, 2 × 6 = 12, 36 + 1 = 37. Triplet (35, 12, 37) and 35² + 12² = 1225 + 144 = 1369 = 37².
- Medium: Number 8. 64 − 1 = 63, 2 × 8 = 16, 64 + 1 = 65. Triplet (63, 16, 65) and 63² + 16² = 3969 + 256 = 4225 = 65².
- Exam type: Number 9. 81 − 1 = 80, 2 × 9 = 18, 81 + 1 = 82. Triplet (80, 18, 82) and 80² + 18² = 6400 + 324 = 6724 = 82².
Pythagorean triplets
In the app you can move this figure with your finger.
Open in the appAll three numbers of a triplet are whole numbers. The square of the largest equals the sum of the squares of the other two.
Accepting a triplet because the sums are close. In (6, 7, 9), 6² + 7² = 36 + 49 = 85 and 9² = 81. 85 and 81 are different, so it is not a triplet.
If you know one triplet, use its multiples. Multiply (6, 8, 10) by 5: 6 × 5 = 30, 8 × 5 = 40, 10 × 5 = 50. And 30² + 40² = 900 + 1600 = 2500 = 50².
Spot the difference
| Point | A triplet | Not a triplet |
|---|---|---|
| Example | 15, 36, 39 | 6, 7, 9 |
| Square of the largest | 1521 | 81 |
| Sum of squares of the other two | 1521 | 85 |
| Result | Both are equal | Both are different |
Practice questions
- 1Is (21, 20, 29) a Pythagorean triplet?
- 2Is (12, 13, 17) a Pythagorean triplet?
- 3Is (30, 72, 78) a Pythagorean triplet?
- 4Is (15, 20, 26) a Pythagorean triplet?
- 5Make a triplet from the number 10 using the rule, and check it.
- 6Write the triplet made from 7 times the numbers (6, 8, 10), and check it.
- 7The sides of a triangle are 12 cm, 35 cm and 37 cm. Is it a right-angled triangle? If so, opposite which side is the right angle?
- 8In a triplet made by multiplying (6, 8, 10), the largest number is 150. Write the triplet and check it.
Show answers (try them yourself first)
- 1The largest is 29. 20² + 21² = 400 + 441 = 841 and 29² = 841; yes. 2) The largest is 17. 12² + 13² = 144 + 169 = 313 and 17² = 289; no. 3) The largest is 78. 30² + 72² = 900 + 5184 = 6084 and 78² = 6084; yes.
- 4The largest is 26. 15² + 20² = 225 + 400 = 625 and 26² = 676; no. 5) 100 − 1 = 99, 2 × 10 = 20, 100 + 1 = 101. 99² + 20² = 9801 + 400 = 10201 = 101²; triplet (99, 20, 101). 6) 6 × 7 = 42, 8 × 7 = 56, 10 × 7 = 70. 42² + 56² = 1764 + 3136 = 4900 = 70²; triplet (42, 56, 70).
- 7The largest is 37. 12² + 35² = 144 + 1225 = 1369 and 37² = 1369; yes, a right-angled triangle. The right angle is opposite the longest side, 37 cm.
- 8150 ÷ 10 = 15, so the multiplier is 15. 6 × 15 = 90, 8 × 15 = 120 and 10 × 15 = 150. 90² + 120² = 8100 + 14400 = 22500 = 150²; triplet (90, 120, 150).
Right-angled triangles in daily life
A curtain, a cupboard corner, a ladder, a pole's wire and a slanting path across a ground all hide a right-angled triangle. One side of it is the hypotenuse, so lengths can be found by the theorem without measuring.
In daily work, first find where the right-angled triangle is. Then decide which side is the hypotenuse. If two right-angle sides are given, add the squares; if the hypotenuse is given, subtract the squares; finally take the square root.
Rule 1: Check from three sides whether a triangle is right-angled — If the sides form a triplet, it is a right-angled triangle. Compare the square of the longest side with the sum of the squares of the other two.
- Easy: Sides 20, 48 and 52. 20² + 48² = 400 + 2304 = 2704 = 52². It is a right-angled triangle.
- Medium: Sides 14, 17 and 22. 14² + 17² = 196 + 289 = 485, but 22² = 484. 485 and 484 are different, so it is not a right-angled triangle.
- Exam type: Sides 1.2, 1.6 and 2. 1.2² + 1.6² = 1.44 + 2.56 = 4 = 2². It is a right-angled triangle.
In the figure: The rectangle is 24 × 32. The diagonal splits the rectangle into two right-angled triangles. The square of the diagonal is 24² + 32² = 576 + 1024 = 1600, so the diagonal is √1600 = 40.
Rule 2: Find the diagonal of a rectangle and of a square — The length and width of a rectangle are the sides forming the right angle. The diagonal is opposite them, so add the squares and take the square root.
- Easy: A rectangle 21 cm × 28 cm. 21² + 28² = 441 + 784 = 1225, diagonal √1225 = 35 cm.
- Medium: A rectangle 45 cm × 60 cm. 45² + 60² = 2025 + 3600 = 5625, diagonal √5625 = 75 cm.
- Exam type: A square of side 10 cm. 10² + 10² = 100 + 100 = 200. 196 < 200 < 225, so 14 < √200 < 15; the diagonal is a little more than 14 cm.
In the figure: A ladder 6.5 metres long leans on a wall and its foot is 2.5 metres from the wall. The ladder is the hypotenuse, so to find the height, 6.5² − 2.5² = 42.25 − 6.25 = 36. The height is √36 = 6 metres.
Rule 3: Ladder, wire and pole — A wall or pole stands at a right angle to the ground. The ladder or wire is the hypotenuse. If the hypotenuse is given, subtract to find a side.
- Easy: A ladder is 2.5 metres long and its foot is 1.5 metres from the wall. 2.5² − 1.5² = 6.25 − 2.25 = 4, height √4 = 2 metres.
- Medium: A wire tied to the top of a pole is 65 metres long and is fixed to the ground 25 metres from the pole. 65² − 25² = 4225 − 625 = 3600, height of the pole √3600 = 60 metres.
- Exam type: Two poles of heights 12 metres and 28 metres stand straight and are 30 metres apart. The difference in heights is 28 − 12 = 16. Distance between the tops: 16² + 30² = 256 + 900 = 1156, √1156 = 34 metres.
In the figure: Someone walked 21 east and then 72 north. Straight distance: 21² + 72² = 441 + 5184 = 5625, so √5625 = 75. The path actually walked is 21 + 72 = 93, which is 93 − 75 = 18 more than the straight distance.
Rule 4: Straight distance after walking in two perpendicular directions — After walking in one direction and then at a right angle to it, the straight distance from the start is the hypotenuse.
- Easy: 30 metres east and 40 metres north. 30² + 40² = 900 + 1600 = 2500, straight distance √2500 = 50 metres. The path walked is 30 + 40 = 70 metres, which is 20 metres more.
- Medium: 48 metres west and 55 metres south. 48² + 55² = 2304 + 3025 = 5329, straight distance √5329 = 73 metres.
- Exam type: 0.7 km east and 2.4 km north. 0.7² + 2.4² = 0.49 + 5.76 = 6.25, straight distance √6.25 = 2.5 km.
Right-angled triangles in daily life
In the app you can move this figure with your finger.
Open in the appAssume the wall or pole is at a right angle to the ground; then the ladder or wire is the hypotenuse. The height is always less than the hypotenuse.
Adding the squares when the ladder is given. 6.5² + 2.5² = 42.25 + 6.25 = 48.5 is wrong; the correct subtraction is 42.25 − 6.25 = 36.
To check the answer: the height 6 is less than the ladder 6.5. Also 2.5² + 6² = 6.25 + 36 = 42.25 = 6.5².
Spot the difference
| Point | Diagonal of a rectangle | Ladder |
|---|---|---|
| Given | 24 and 32 | 6.5 and 2.5 |
| Operation on squares | Addition | Subtraction |
| Square in the figure | 1600 | 36 |
| Answer | 40 | 6 |
Practice questions
- 1Is a triangle with sides 16, 30 and 34 right-angled?
- 2A rectangle is 18 cm × 24 cm. Find its diagonal.
- 3A ladder 10 metres long has its foot 6 metres from the wall. How high up the wall does it reach?
- 4Is a triangle with sides 10, 11 and 14 right-angled?
- 5A square ground has a side of 60 metres. If you walk along the diagonal, between which two whole numbers does the distance lie?
- 6A wire 50 metres long is tied to the top of a pole and fixed to the ground 14 metres from the foot of the pole. What is the height of the pole?
- 7Two friends start from the same place. One walks 36 metres east and the other 15 metres north. How far apart are they now?
- 8The diagonal of a rectangle is 100 cm and its length is 80 cm. Find the width, the perimeter and the area.
Show answers (try them yourself first)
- 1The largest is 34. 16² + 30² = 256 + 900 = 1156 and 34² = 1156; yes, right-angled. 2) 18² + 24² = 324 + 576 = 900, diagonal √900 = 30 cm. 3) 10² − 6² = 100 − 36 = 64, height √64 = 8 metres.
- 4The largest is 14. 10² + 11² = 100 + 121 = 221 and 14² = 196; no. 5) 60² + 60² = 3600 + 3600 = 7200. 84² = 7056 and 85² = 7225; 7056 < 7200 < 7225, so 84 < √7200 < 85. 6) 50² − 14² = 2500 − 196 = 2304, height √2304 = 48 metres.
- 736² + 15² = 1296 + 225 = 1521, distance √1521 = 39 metres.
- 8100² − 80² = 10000 − 6400 = 3600, width √3600 = 60 cm. Perimeter 2 × (80 + 60) = 280 cm. Area 80 × 60 = 4800 sq cm.