SHSAT: Square roots and the Pythagorean theorem
Find missing sides of right triangles, distances between points, and estimates or simplified forms of square roots.
How do you find a missing side of a right triangle or the distance between two points without a calculator?
Short answer
Use a² + b² = c², where c is the hypotenuse. Add squares to find the hypotenuse, subtract to find a leg, and take the square root at the end.
Know first: Squaring numbers, Perfect squares up to 400
Step 1 of 5
In a right triangle, the squares of the legs add to the square of the hypotenuse
The Pythagorean theorem says that in a right triangle with legs a and b and hypotenuse c, a² + b² = c². The hypotenuse is the longest side, across from the right angle.
To find the hypotenuse, add the squares of the legs. To find a leg, subtract: b² = c² − a². Either way, take the square root at the end.
| Sides | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 7, 24, 25 | 49 + 576 = 625 |
| 9, 40, 41 | 81 + 1600 = 1681 |
| any multiple, such as 10, 24, 26 | 100 + 576 = 676 |
Recognizing these saves time, since the SHSAT has no calculator.
Step 2 of 5
Distance is the hypotenuse of a right triangle
Worked example: Distance between two points
What is the distance between (−6, 2) and (14, 23)?
- A1
- B29
- C41
- D841
- 1
Horizontal leg
14 − (−6) = 20.
- 2
Vertical leg
23 − 2 = 21.
- 3
Apply the theorem
20² + 21² = 400 + 441 = 841, so the distance is √841 = 29, because 29² = 841.
- 4
See why the other choices are there
1 subtracts the legs. 41 adds the legs, which is the walking distance along the grid, not the straight line. 841 stops before taking the square root.
Answer
29
Find the wrong step
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
- 1
Write a² + b² = c² with c = 13.
- 2
b² = 13² + 5² = 194.
What went wrong
The hypotenuse is the longest side. To find a leg, subtract the known leg's square from the hypotenuse's square.
- 3
b = √194.
The fix
b² = 169 − 25 = 144, so the leg is 12.
Step 3 of 5
Square, add or subtract, then root
Check yourself · Question 1
Between which two consecutive integers does √58 lie?
Answer: B
7² = 49 and 8² = 64. Since 49 < 58 < 64, √58 is between 7 and 8.
Check yourself · Question 2
A rectangular park is 12 meters by 35 meters. How many meters shorter is the diagonal path than walking along two adjacent edges?
Answer: A
Diagonal: 12² + 35² = 144 + 1225 = 1369, and √1369 = 37. The edges total 47. The diagonal saves 47 − 37 = 10 meters.
Trap answer: 37
Stopping one step early
- Why it looks right
- Finding the diagonal is the hard part, and 37 is a satisfying whole number.
- Why it's wrong
- The question asks how much shorter the diagonal is. You still need to compare it with the 47-meter walk.
- The right answer
- 10 meters, from 47 − 37.
Check yourself · Question 3
A square has an area of 50 square centimeters. What is its perimeter in centimeters?
Answer: C
Side = √50 = √(25 × 2) = 5√2. Perimeter = 4 × 5√2 = 20√2.
Common mistake
I take the square root of each square separately: √(a² + b²) = a + b.
- Why it's tempting
- Square roots and squares seem to undo each other, so you cancel them term by term.
- Do this instead
- Add the squares first, then take one square root. √(12² + 16²) = √400 = 20, not 28.
Step 4 of 5
What to remember
Remember
- a² + b² = c², with c the hypotenuse across from the right angle. Subtract to find a missing leg.
- The distance between two points is the hypotenuse of a right triangle with legs equal to the coordinate differences.
- Estimate roots between perfect squares, and remember that a root doesn't split over a sum.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which side was the hypotenuse, and what did you get when you added the squares of the legs?