We are given five right triangles, and in each case, we need to find the length of the side marked
x using the
Pythagorean Theorem.
The Pythagorean Theorem states:
$$
a^2 + b^2 = c^2
$$
Where:
- $ a $ and $ b $ are the legs (the two shorter sides),
- $ c $ is the hypotenuse (the longest side, opposite the right angle).
We'll solve each problem one by one.
---
1.
Given:
- One leg = 15 cm
- Other leg = 8 cm
- Hypotenuse = $ x $
Apply the theorem:
$$
x^2 = 15^2 + 8^2 = 225 + 64 = 289
$$
$$
x = \sqrt{289} = 17
$$
✔ Answer: $ x = 17 $ cm
---
2.
Given:
- One leg = 9 cm
- Hypotenuse = 20 cm
- Other leg = $ x $
$$
x^2 + 9^2 = 20^2
\Rightarrow x^2 + 81 = 400
\Rightarrow x^2 = 400 - 81 = 319
$$
$$
x = \sqrt{319} \approx 17.86 \text{ cm}
$$
But let's keep it exact unless asked for decimal.
✔ Answer: $ x = \sqrt{319} \approx 17.86 $ cm
(If rounded to nearest tenth:
17.9 cm)
---
3.
Given:
- One leg = 6.5 cm
- Other leg = 10.2 cm
- Hypotenuse = $ x $
$$
x^2 = 6.5^2 + 10.2^2 = 42.25 + 104.04 = 146.29
$$
$$
x = \sqrt{146.29} \approx 12.1 \text{ cm}
$$
✔ Answer: $ x \approx 12.1 $ cm
---
4.
Given:
- One leg = 10.5 cm
- Other leg = 8.4 cm
- Hypotenuse = $ x $
Wait — actually, looking at the diagram:
- The vertical leg is 8.4 cm
- The horizontal leg is $ x $
- The hypotenuse is 10.5 cm
So this is a right triangle where:
- $ x $ is a leg,
- 8.4 cm is another leg,
- 10.5 cm is the hypotenuse.
So:
$$
x^2 + 8.4^2 = 10.5^2
\Rightarrow x^2 + 70.56 = 110.25
\Rightarrow x^2 = 110.25 - 70.56 = 39.69
$$
$$
x = \sqrt{39.69} = 6.3 \text{ cm}
$$
✔ Answer: $ x = 6.3 $ cm
---
5.
Given:
- One leg = 5 cm
- Hypotenuse = 13 cm
- Other leg = $ x $
$$
x^2 + 5^2 = 13^2
\Rightarrow x^2 + 25 = 169
\Rightarrow x^2 = 144
\Rightarrow x = \sqrt{144} = 12
$$
✔ Answer: $ x = 12 $ cm
---
✔ Final Answers:
1. $ x = 17 $ cm
2. $ x = \sqrt{319} \approx 17.9 $ cm
3. $ x \approx 12.1 $ cm
4. $ x = 6.3 $ cm
5. $ x = 12 $ cm
Let me know if you want exact values or rounded answers!
Parent Tip: Review the logic above to help your child master the concept of basic pythagorean theorem worksheet.
