This MCQ module is based on: Triangle Basics and Equilateral Triangles
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Triangle Basics and Equilateral Triangles
🎓 Class 7
Mathematics
CBSE
Theory
Ch 7 — A Tale of Three Intersecting Lines
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Triangle Basics and Equilateral Triangles
Targeting Class 7 level in Geometry, with Basic difficulty.
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Chapter 7 — A Tale of Three Intersecting Lines
A triangle? is the simplest closed figure. It has three corner points called vertices, and three line segments joining pairs of vertices called sides. Triangles come in many shapes and sizes.
Three different triangles. We write them as \(\triangle ABC\), \(\triangle XYZ\), \(\triangle WMU\).
The symbol \(\triangle\) stands for "triangle". When naming a triangle, the three letters are written in the order the vertices appear. The three angles \(\angle CAB,\ \angle ABC,\ \angle BCA\) are often shortened to \(\angle A,\ \angle B,\ \angle C\).
Question. What happens when the three vertices of a triangle lie on a straight line? Then the figure collapses — it is no longer a triangle; we just get a line segment.
7.1 Equilateral Triangles
Among all triangles, the equilateral triangle is the most symmetric. All three sides have equal length. Let's try to build one.
Construct a triangle in which all sides are 4 cm long.
We cannot succeed using only a marked ruler and pencil — we would need many trials. But with a compass, the construction works perfectly the first time.
Step 1. Draw segment AB of length 4 cm.
Step 2. With centre A and radius 4 cm, draw an arc. The third vertex C must lie somewhere on this arc (every point on it is 4 cm from A).
Step 3. With centre B and radius 4 cm, draw another arc. Every point on this arc is 4 cm from B.
Step 4. Let C be the point where the two arcs meet. Both AC and BC are then 4 cm.
Step 5. Join AC and BC to complete \(\triangle ABC\).
Fig 7.1 — Two arcs of radius 4 cm meet at C, forming equilateral \(\triangle ABC\).
Why it works
Point C lies on both circles. Being on the first circle means \(AC = 4\) cm. Being on the second circle means \(BC = 4\) cm. So all three sides equal 4 cm.Isosceles Triangles
A triangle in which at least two sides have equal length is called an isosceles triangle?. Every equilateral triangle is also isosceles (three equal sides include every pair). The equal sides are marked with identical tick marks.
Isosceles \(\triangle PQR\) with PR = QR (equal sides marked).
Figure it Out
Q1. Construct triangles having the following side lengths (in cm): (a) 4, 4, 6 (b) 3, 4, 5 (c) 1, 5, 5 (d) 4, 6, 8 (e) 3.5, 3.5, 3.5. Which of these are equilateral?
Only (e) 3.5, 3.5, 3.5 is equilateral (all three sides equal). (a), (c) are isosceles. (b) and (d) are scalene.
Q2. Use points on a single circle (or the centre and a point on the circle) to form isosceles triangles.
Pick the centre O and any two points A, B on the circle. Since OA and OB are both radii, they are equal, so \(\triangle OAB\) is isosceles. Many such triangles can be drawn.
Activity: The Two-Arc Game
Materials: Compass, ruler, pencil, plain paper.
- Draw segment AB of length 5 cm.
- Set compass to 5 cm; draw arcs centred at A and at B above AB.
- Mark C where the arcs cross. Join AC and BC.
- Measure AC and BC with the ruler. What do you notice?
- Try again with arcs of 6 cm, 7 cm. Predict where C will be.
AC = BC = the arc radius, so the triangle is always equilateral. As the radius grows, C rises higher above AB (the peak is at height \(\frac{\sqrt{3}}{2}\times\text{radius}\)).
Competency-Based Questions
Scenario: Asha is decorating a banner with small equilateral triangles cut from coloured paper. She has a compass and a ruler. She wants every triangle to have sides of 6 cm.
Q1. Which compass radius should Asha use when drawing the two arcs to locate the third vertex?
L3 Apply6 cm — the radius must equal the desired side length so that both AC and BC come out as 6 cm.
Q2. Asha claims: "If I set the compass to 3 cm and repeat the construction on a base AB of 6 cm, I will still get a triangle." Analyse whether she is correct.
L4 AnalyseIncorrect. Two arcs of 3 cm centred at A and at B (which are 6 cm apart) would just touch at the midpoint of AB — they do not cross above the line. No triangle is formed because \(3+3=6\), not greater than 6.
Q3. Evaluate whether the following three lengths can form an equilateral triangle: 7 cm, 7 cm, 7.1 cm. Justify.
L5 EvaluateNo — all three sides must be equal. 7, 7, 7.1 gives an isosceles (almost equilateral) triangle, not an equilateral one.
Q4. Design a step-by-step procedure to construct an isosceles (not equilateral) triangle with two sides of 5 cm and a base of 8 cm.
L6 CreateStep 1: Draw AB = 8 cm. Step 2: With centre A, draw an arc of radius 5 cm above AB. Step 3: With centre B, draw an arc of radius 5 cm above AB. Step 4: Mark C where the arcs intersect. Step 5: Join AC and BC. Then AC = BC = 5 cm, AB = 8 cm — an isosceles triangle.
Assertion–Reason Questions
Assertion (A): An equilateral triangle is also an isosceles triangle.
Reason (R): An isosceles triangle requires only at least two sides to be equal.
Reason (R): An isosceles triangle requires only at least two sides to be equal.
(a) — all three sides equal ⇒ in particular two are equal ⇒ isosceles. R explains A.
Assertion (A): Every point on a circle of radius 4 cm centred at A is 4 cm away from A.
Reason (R): The radius of a circle is the distance from the centre to any point on the circle.
Reason (R): The radius of a circle is the distance from the centre to any point on the circle.
(a) — this is exactly the definition of a circle, so R explains A.
Frequently Asked Questions
What are the three types of triangles by sides?
Equilateral (all three sides equal), isosceles (exactly two sides equal) and scalene (all three sides different). Side length directly controls the interior angles.
Why are all angles of an equilateral triangle 60 degrees?
Because the interior angles sum to 180 degrees and all three angles are equal due to the equal sides, each angle must be 180 divided by 3, which is 60 degrees.
What are the vertices of a triangle?
The vertices are the three corner points where two sides meet. A triangle labelled ABC has vertices A, B and C, with opposite sides commonly denoted a, b and c.
How do you draw an equilateral triangle with compass and ruler?
Draw a base segment AB of the required length. Place the compass at A with that length as radius and draw an arc. Do the same from B. The two arcs meet at C. Join AC and BC.
What real objects are triangular?
Traffic signs, roof trusses, bridge girders, musical triangles, Pyramid faces and many logos use triangles for their rigidity and visual balance.
AI Tutor
Mathematics Class 7 — Ganita Prakash
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