Pythagorean Theorem practice worksheet with real-world and geometric problems.

Here are the step-by-step solutions for each question on your worksheet.

Question 1
This problem involves a ladder leaning against a wall, which forms a right-angled triangle.
* The ladder is the longest side (hypotenuse), so $c = 9$.
* The distance from the wall is one of the shorter sides, so $a = 1.5$.
* We need to find how high up the wall the ladder reaches ($b$).

Using the formula $a^2 + b^2 = c^2$:
1. Square the known numbers: $1.5^2 = 2.25$ and $9^2 = 81$.
2. Set up the equation: $2.25 + b^2 = 81$.
3. Subtract 2.25 from 81 to isolate $b^2$: $81 - 2.25 = 78.75$.
4. Find the square root of 78.75 to get $b$.
$\sqrt{78.75} \approx 8.87$ meters.

Question 2
A diagonal cuts a square into two right-angled triangles. The sides of the square are the legs ($a$ and $b$), and the diagonal is the hypotenuse ($x$).
* Side length $a = 5$ and $b = 5$.
* Formula: $5^2 + 5^2 = x^2$.
1. Calculate squares: $25 + 25 = 50$.
2. So, $x^2 = 50$.
3. Find the square root of 50.
$\sqrt{50} \approx 7.07$ cm.

Question 3
We have a right-angled triangle with a hypotenuse of 25 cm and one leg of 7 cm. First, we must find the missing leg (let's call it $a$) to solve parts (a) and (b).

* Find the missing side:
$a^2 + 7^2 = 25^2$
$a^2 + 49 = 625$
$a^2 = 625 - 49$
$a^2 = 576$
$a = \sqrt{576} = 24$ cm.
So, the three sides are 7 cm, 24 cm, and 25 cm.

* (a) Perimeter: Add all three sides together.
$7 + 24 + 25 = 56$ cm.

* (b) Area: Use the formula $\frac{1}{2} \times \text{base} \times \text{height}$. The base and height are the two shorter sides (legs).
$\text{Area} = \frac{1}{2} \times 7 \times 24$
$\text{Area} = 7 \times 12 = 84$ cm².

Question 4
The diagonal of a rectangle creates a right-angled triangle with the length and width as the legs.
* Length ($a$) = 20 cm, Width ($b$) = 8 cm.
* Formula: $20^2 + 8^2 = \text{diagonal}^2$.
1. Calculate squares: $400 + 64 = 464$.
2. Find the square root of 464.
$\sqrt{464} \approx 21.54$ cm.

──────────────────────────────────────

Final Answer:
Question 1: 8.87 m
Question 2: 7.07 cm
Question 3: (a) 56 cm, (b) 84 cm²
Question 4: 21.54 cm