Pythagoras in Right Triangles

What you’ll learn

Find a missing side only when a triangle has a right angle.

រកជ្រុងមិនស្គាល់ក្នុងត្រីកោណកែង។

Meaning and use

Pythagoras links the side lengths of a right triangle. The hypotenuse is opposite the right angle and is the longest side. To find a shorter side, subtract squares rather than adding them. Take the positive root for a physical length.

ទ្រឹស្ដីបទពីតាគ័រប្រើសម្រាប់ត្រីកោណកែង។ អ៊ីប៉ូតេនុសនៅទល់មុខមុំកែង និងវែងបំផុត។ រកជ្រុងខ្លីត្រូវដកការ៉េ រួចយកឫសវិជ្ជមាន។

Formulas

Right-triangle relation

\[a^2+b^2=c^2\]

Missing shorter side

\[a=\sqrt{c^2-b^2}\]

Symbols and units

Symbol	Meaning	Unit or restriction
a, b	Sides enclosing the right angle	Same length unit
c	Hypotenuse, opposite the right angle	Same length unit

Conditions and limits

• The triangle must have a 90° angle.
• Use the hypotenuse in the c position and ensure c is longer than either other side.
• លទ្ធផលប្រវែងយកឫសវិជ្ជមាន។

Worked examples

Finding the hypotenuse

The perpendicular sides are 3 cm and 4 cm.

1. Calculate \(c^2=3^2+4^2=25\).
2. Take \(c=\sqrt{25}=5\) cm.

5 cm

ត្រូវបូកការ៉េ រួចយកឫស មិនមែនបូកប្រវែងដោយផ្ទាល់។

Finding a shorter side

Hypotenuse is 13 m and one shorter side is 5 m.

1. Calculate \(a^2=13^2-5^2=144\).
2. Take \(a=\sqrt{144}=12\) m.

12 m

ពេលរកជ្រុងខ្លី ត្រូវដកការ៉េពីការ៉េអ៊ីប៉ូតេនុស។

Common mistakes

Instead of	Use	Why
A 3 cm and 4 cm right triangle has hypotenuse 7 cm.	Its hypotenuse is 5 cm.	Add squares and then take a square root.
Use a² + b² = c² for every triangle.	Use it only for a right triangle.	The right-angle condition is essential.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Find the hypotenuse for perpendicular sides 6 and 8 cm.
2. Find a shorter side if c = 10 cm and the other side is 6 cm.
3. Which side is c?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 10 cm

   √(36 + 64) = 10.

2. 8 cm

   √(100 − 36) = 8.

3. The side opposite the right angle.

   It is the hypotenuse.

Quick recap

• Identify the right angle first.
• Label the hypotenuse correctly.
• Take the square root to recover a length.