48 Pythagorean Theorem Worksheet with Answers [Word + PDF] - Free Printable

Here are the step-by-step solutions for the three problems on your worksheet.

Q1) Isosceles Triangle

Goal: Find the perpendicular distance (height) from point A to the base BC.

1. Understand the shape: We have an isosceles triangle where side $AB = 12$ cm and side $AC = 12$ cm. The base $BC$ is $10$ cm.
2. Split the triangle: When you draw a perpendicular line from the top vertex (A) to the base (BC) in an isosceles triangle, it cuts the base exactly in half.
* Total base $BC = 10$ cm.
* Half of the base $= 10 \div 2 = 5$ cm.
3. Use Pythagoras' Theorem: Now we have a right-angled triangle with:
* Hypotenuse (side AB) $= 12$ cm
* Base $= 5$ cm
* Height ($h$) $= ?$

The formula is: $a^2 + b^2 = c^2$
$$h^2 + 5^2 = 12^2$$
4. Calculate:
* $5^2 = 25$
* $12^2 = 144$
* So, $h^2 + 25 = 144$
* Subtract 25 from both sides: $h^2 = 144 - 25$
* $h^2 = 119$
* To find $h$, take the square root: $h = \sqrt{119}$
* $\sqrt{119} \approx 10.91$ cm

Q2) Distance between Coordinates

Goal: Find the length of line AB given coordinates $A(4, 7)$ and $B(16, 12)$.

1. Find the horizontal distance (change in x):
* Subtract the x-coordinates: $16 - 4 = 12$.
2. Find the vertical distance (change in y):
* Subtract the y-coordinates: $12 - 7 = 5$.
3. Use Pythagoras' Theorem: Imagine a right-angled triangle where the horizontal side is 12 and the vertical side is 5. The line AB is the hypotenuse.
* $Length^2 = 12^2 + 5^2$
4. Calculate:
* $12^2 = 144$
* $5^2 = 25$
* $Length^2 = 144 + 25$
* $Length^2 = 169$
* Take the square root: $Length = \sqrt{169}$
* $Length = 13$

Q3) The Ladder Problem

Goal: Calculate the height of the windowsill.

1. Identify the sides:
* The ladder is the hypotenuse (longest side) $= 8$ m.
* The distance from the wall is the base $= 6$ m.
* The height of the windowsill is the unknown side ($h$).
2. Use Pythagoras' Theorem:
* $h^2 + 6^2 = 8^2$
3. Calculate:
* $6^2 = 36$
* $8^2 = 64$
* So, $h^2 + 36 = 64$
* Subtract 36 from both sides: $h^2 = 64 - 36$
* $h^2 = 28$
* To find $h$, take the square root: $h = \sqrt{28}$
* $\sqrt{28} \approx 5.29$ m

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Final Answer:
Q1) 10.91 cm (or $\sqrt{119}$ cm)
Q2) 13 units
Q3) 5.29 m (or $\sqrt{28}$ m)