Mathematics NCERT Class 7 Ch-6 The triangle and its properties

Mathematics NCERT Class 7

Ch-6 The triangle and its properties

by Jil Parmar

​INTRODUCTION

A triangle is a simple closed figure made up of three line segments.

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In this triangle ABC ,

The sides are - AB, BC and CA

Vertices are - A, B, C

Angles are - ABC , BCA AND CAB

​ Triangles are classified on the basis of (i) sides (ii) angles.

(i)                Based on Sides: Scalene, Isosceles and Equilateral triangles.

(ii)                Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.

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Open Ended

1.     Write the:

(i)                 Side opposite to the vertex Q of DPQR

(ii)                Angle opposite to the side LM of DLMN

(iii)                Vertex opposite to the side RT of DRST

Open Ended

The six elements of a triangle are its 3 sides and 3 angles. Write the six elements of triangle ABC

Open Ended

Here are a few questions from NCERT Book .

Q. 1.     Look at Fig 6.2 and classify each of the triangles according to its

(a)           Sides

(b)            Angles

Medians of a triangle

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The line segment which connects the vertex to the midpoint of the opposite side is called a median.

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Multiple Choice

How many medians can a triangle have ?

• 5

• 3

• 1

• 2

Multiple Choice

Does median lie wholly in the interior of a triangle ?

• Yes

• No

• Maybe

• Cannot be determined

​Altitudes of a triangle

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Altitude means height.

​A perpendicular line which connects the vertex to the opposite side forming a 90 degree is called an altitude.

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Multiple Choice

How many altitudes can a triangle have?

• Infinite

• 10

• 3

• 1

Multiple Choice

In which type of triangle two of its altitudes are tow of its sides?

• Right angled

• obtuse angled

• acute angled

• complete angled

Fill in the Blank

In which type of triangle does the altitude lie in exterior ?

Draw

Draw rough sketches of altitudes for the following triangles.

Open Ended

In Δ PQR, D is the mid-point of QR

PM is _________________.

PD is _________________.

Is QM = MR?

Draw

Draw rough sketches for the following: (a) In ΔABC, BE is a median. (b) In ΔPQR, PQ and PR are altitudes of the triangle. (c) In ΔXYZ, YL is an altitude in the exterior of the triangle.

Draw

Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same

​EXTERIOR ANGLE PROPERTY OF A TRIANGLE

​An exterior angle of a triangle is equal to the sum of its interior opposite angles.

​See the given example

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Open Ended

. Are the exterior angles formed at each vertex of a triangle equal?

Open Ended

Find the unknown value of X

Open Ended

Find the unknown values in each of them.

​Angle sum property of a triangle

​The total measure of the three angles of a triangle is 180°.

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Open Ended

Find the unknown values of each of the following questions.

Open Ended

1. Two angles of a triangle are 30º and 80º. Find the third angle.

2. One of the angles of a triangle is 80º and the other two angles are equal. Find the measure of each of the equal angles.

3. The three angles of a triangle are in the ratio 1:2:1. Find all the angles of the triangle. Classify the triangle in two different ways.

Open Ended

1. Can you have a triangle with two right angles?

2. Can you have a triangle with two obtuse angles?

3. Can you have a triangle with two acute angles?

4. Can you have a triangle with all the three angles greater than 60º?

5. Can you have a triangle with all the three angles equal to 60º?

6. Can you have a triangle with all the three angles less than 60º?

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​TWO SPECIAL TRIANGLES : EQUILATERAL AND ISOSCELES

A triangle in which all the three sides are of equal lengths is called an equilateral triangle.

​A triangle in which two sides are of equal lengths is called an isosceles triangle.

Thus, in an isosceles triangle: (i) two sides have same length. (ii) base angles opposite to the equal sides are equal.

Open Ended

Find the unknown value in each case.

​SUM AND DIFFERENCE OF LENGTHS PROPERTY

​the sum of the lengths of any two sides of a triangle is greater than the third side

​the difference of lengths of any two sides is lesser than the third side

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​EXAMPLE Is there a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm?

SOLUTION

Suppose such a triangle is possible. Then the sum of the lengths of any two sides would be greater than the length of the third side.

Let us check this.

Is 4.5 + 5.8 >10.2? Yes

Is 5.8 + 10.2 > 4.5? Yes

Is 10.2 + 4.5 > 5.8? Yes

Therefore, the triangle is possible.

​EXAMPLE The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two numbers can length of the third side fall?

SOLUTION

We know that the sum of two sides of a triangle is always greater than the third.

Therefore, third side has to be less than the sum of the two sides.

The third side is thus, less than 8 + 6 = 14 cm.

The side cannot be less than the difference of the two sides.

Thus, the third side has to be more than 8 – 6 = 2 cm.

The length of the third side could be any length greater than 2 and less than 14 cm.

Open Ended

. Is it possible to have a triangle with the following sides?

(i) 2 cm, 3 cm, 5 cm

(ii) 3 cm, 6 cm, 7 cm

(iii) 6 cm, 3 cm, 2 cm

Open Ended

Take any point O in the interior of a triangle PQR.

Is (i) OP + OQ > PQ?

(ii) OQ + OR > QR?

(iii) OR + OP > RP?

Open Ended

AM is a median of a triangle ABC.

1. Is AB + BC + CA > 2 AM? (Consider the sides of triangles ΔABM and ΔAMC.)

2. ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

Open Ended

The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?

​PTHAGORAS PROPERTY

​Pythagoras, a Greek philosopher of sixth century B.C. is said to have found a very important and useful property of right-angled triangles given in this section. The property is, hence, named after him. In fact, this property was known to people of many other countries too. The Indian mathematician Baudhayan has also given an equivalent form of this property. We now try to explain the Pythagoras property. In a right-angled triangle, the sides have some special names. The side opposite to the right angle is called the hypotenuse; the other two sides are known as the legs of the right-angled triangle.

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​In a right-angled triangle, the square on the hypotenuse = sum of the squares on the legs.

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EXAMPLE 5 Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is a right-angled triangle. SOLUTION 32 = 3 × 3 = 9; 42 = 4 × 4 = 16; 52 = 5 × 5 = 25 We find 32 + 42 = 52 . Therefore, the triangle is right-angled. Note: In any right-angled triangle, the hypotenuse happens to be the longest side. In this example, the side with length 5 cm is the hypotenuse.

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​EXAMPLE 6 Δ ABC is right-angled at C. If AC = 5 cm and BC = 12 cm find the length of AB.

SOLUTION

By Pythagoras property, AB2 = AC2 + BC2

= 52 + 122

= 25 + 144 = 169 = 132 or AB2 = 132 .

So, AB = 13 or the length of AB is 13 cm.

Note: To identify perfect squares, you may use prime factorisation technique.

Open Ended

find the value of x

Open Ended

A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.

Open Ended

4. Which of the following can be the sides of a right triangle? (i) 2.5 cm,6.5 cm, 6 cm. (ii) 2 cm, 2 cm, 5 cm. (iii) 1.5 cm, 2cm, 2.5 cm. In the case of right-angled triangles, identify the right angles.

Open Ended

5. A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.

Open Ended

8. The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.