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Understanding Elementary Shapes - Perpendicular Lines

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Perpendicular lines are two lines that intersect each other such that the angle between them is a right angle (90∘90^{\circ}). Visually, these lines look like the intersection of the letter 'T' or the edges of a square meeting at a corner.

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The symbol used to represent perpendicular lines is ⊥\perp. For example, if a line ABAB is perpendicular to line CDCD, it is written mathematically as AB⊥CDAB \perp CD.

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When two lines are perpendicular, all four angles formed at the point of intersection are equal to 90∘90^{\circ}. This creates a perfect 'plus' (++) shape where each quadrant is identical in angle measure.

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A perpendicular bisector is a line that is perpendicular to a given line segment and passes through its midpoint, dividing the segment into two equal lengths. Visualizing a horizontal segment ABAB being cut exactly in the middle by a vertical line XYXY represents a perpendicular bisector.

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In geometric tools, 'Set-Squares' are used to draw perpendicular lines. These are triangular tools where one angle is exactly 90∘90^{\circ}, allowing students to align one edge with a line and draw the perpendicular line along the other edge.

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Perpendicular lines are frequently seen in real-world objects. For instance, the adjacent edges of a postcard, the vertical pole of a lamp post meeting the horizontal ground, and the corners of a rectangular window all represent perpendicularity.

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If two lines are perpendicular to the same line in a plane, they are parallel to each other. Visualizing the two vertical sides of a ladder being perpendicular to each horizontal rung shows how the vertical sides stay parallel.

📐Formulae

If L1⊥L2, then the angle θ=90∘\text{If } L_1 \perp L_2, \text{ then the angle } \theta = 90^{\circ}

For a perpendicular bisector of segment AB at midpoint M:AM=MB=12AB\text{For a perpendicular bisector of segment } AB \text{ at midpoint } M: AM = MB = \frac{1}{2} AB

Area of a Right-Angled Triangle (formed by perpendicular base and height)=12×base×height\text{Area of a Right-Angled Triangle (formed by perpendicular base and height)} = \frac{1}{2} \times \text{base} \times \text{height}

💡Examples

Problem 1:

A line segment PQPQ of length 10 cm10 \text{ cm} has a perpendicular bisector XYXY that intersects PQPQ at point OO. Find the length of POPO.

Solution:

Step 1: Identify the properties of a perpendicular bisector. A perpendicular bisector divides a line segment into two equal parts at a 90∘90^{\circ} angle. Step 2: Since XYXY is the perpendicular bisector of PQPQ at OO, point OO is the midpoint of PQPQ. Step 3: Use the midpoint formula: PO=12×PQPO = \frac{1}{2} \times PQ. Step 4: Substitute the given value: PO=12×10 cm=5 cmPO = \frac{1}{2} \times 10 \text{ cm} = 5 \text{ cm}.

Explanation:

The definition of a bisector ensures that the segment is split into two equal halves, so we simply divide the total length by 22.

Problem 2:

In the English alphabet 'L', if the vertical bar is line segment ABAB and the horizontal bar is BCBC, identify the relationship between ABAB and BCBC and state the measure of ∠ABC\angle ABC.

Solution:

Step 1: Observe the shape of the letter 'L'. The two segments meet at a sharp corner. Step 2: In geometry, the edges of an 'L' shape are perpendicular to each other. Step 3: Write the relationship using the symbol: AB⊥BCAB \perp BC. Step 4: Since the lines are perpendicular, the angle formed at the vertex BB is a right angle. Step 5: Therefore, ∠ABC=90∘\angle ABC = 90^{\circ}.

Explanation:

Perpendicularity is defined by the intersection of lines at a 90∘90^{\circ} angle, which is the characteristic shape of the letter 'L'.