Class 7 Maths Chapter 6 Question Answers - The Triangle and Its Properties

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Q1. In a triangle ABC, let D be a point on the segment BC such that AB + BD = AC + CD. Suppose that the points B,C, and the centroids of ΔABD & ΔACD lie on a circle. Prove that AB=AC.
Ans:
Let G₁ ,G₂ denote the centroids of triangles ABD and ACD.
Then G₁ ,G₂  lie on the line parallel to BC that passes through the centriod of triangle ABC.
∴ BG₁ G₂ C is an isosceles trapezoid.
Therefore it follows that BG₁ =CG₂ .
∴ AB² + BD² = AC² +CD² .
Hence it follows that AB⋅BD = AC⋅CD.
Therefore the sets {AB, BD} and {AC, CD} are the same (since they are both equal to the set of roots of the same polynomial).
Note that if AB = CD then AC = BD and then AB + AC = BC, a contradiction.
∴ AB = AC.

Q2. Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, and 5 cm is a right angled triangle?
Ans:
The given lengths of the sides are 3 cm,4 cm and 5 cm
3² = 3 × 3 = 9;
4²  = 4 × 4 = 16
5² = 5 × 5 = 25
We know that
9 + 16 = 25
3² + 4²
= 5²
Therefore, the triangle is a right-angled triangle.
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.

Q3. The base of a right prism is a right angled triangle. The measure of the base of the right angled triangle is 3 m and its height 4 m. If the height of the prism is 7 m then find
(i) the number of edges of the prism
(ii) the volume of the prism
(iii) the total surface area of the prism.
Ans:
(i) The number of the edges = The number of sides of the base x 3 = 3  x  3 = 9
(ii) The volume of the prism  = Area of the base x Height of the prism = 1/2(3 x 4) x 7
= 42m³
(iii) TSA = LSA + 2(area of base) = ph + 2(area of base) where p = perimeter of the base = sum of length of the sides of the given triangle
As hypotenuse of the triangle  = √3² + 4²
= √25
= 5 m

Q4. Write the following statement in five different ways conveying the same meaning p : If triangle is equiangular then it is an obtuse angled triangle
Ans:
The given statement can be written in five different ways as follows,
(i) A triangle is equiangular implies that it is an obtuse-angled triangle.
(ii) A triangle is equiangular only if it is an obtuse-angled triangle.
(iii) For a triangle to be equiangular it is necessary that the triangle is an obtuse-angled triangle.
(iv) For a triangle to be an obtuse-angled triangle it is sufficient that the triangle is equiangular.
(v) If a triangle is not an obtuse-angled triangle then the triangle is not equiangular.

Q5. The angles of a triangle are in the ratio of 2 : 3 : 4. What is the measure of the smallest interior angle of the triangle?
Ans:
By angle sum property, the sum of angles is 180^o .
Let 2x, 3x and 4x be the three angles.
2x + 3x + 4x = 180^o
9x = 180^o
x = 20^o
Smallest interior angle = 2 × 20^o
= 40^o .

Q.6. The angles of a triangle are in the ratio of 3 : 5 : 7. What is the measure of the largest interior angle of the triangle?
Ans:
By angle sum property, the sum of angles is 180^o .
Let 3x,5x and 7x be the three angles of the triangle.
So, 3x + 5x + 7x = 180^o
15x = 180^o
x = 12^o
Hence, Largest interior angle = 7 × 12^o
= 84^o

Q7. Using converse of Pythagoras theorem, check whether the sides form a right angled triangle 13,15 and 16?
Ans:
According to the Pythagoras theorem-In a right angle triangle square of the hypotenuse is equals to the sum of the square of the other two sides.
⇒ a² + b² = c²
∴ 13² +15² =16²
⇒ 13² +15²
⇒ 169 + 225 = 394
⇒ 16²= 256
∴ 394 ≠ =256
Hence, the given sides does not form a right angled triangle.

Q8. Using converse of Pythagoras theorem, check whether the sides form a right angled triangle 63,65 and 16?
Ans:
According to the Pythagoras theorem-In a right angle triangle square of the hypotenuse is equals to the sum of the square of the other two sides.
⇒ a² + b² = c²
∴ 63² +16² = 65²
⇒ 63² +16²
⇒ 3969 + 256 = 4225
⇒ 65² = 4225
∴4225 = 4225
Hence, the given sides  form a right angled triangle.

Q9. Which side is the hypotenuse for the sides, 48,73 and 55 in a right angled ? (Apply Converse of Pythagoras theorem).
Ans:
According to the Pythagoras theorem, In a right angle triangle square of the hypotenuse equals the sum of the square of the other two sides.
⇒ a² +b² = c²
∴ 48² + 55² = 73²
⇒ 48² + 55² ⇒ 2304 + 3025 = 5329
Also, ⇒ 73²  =5329
∴ LHS = RHS = 5329
Hence the given sides form a right-angled triangle.
As the longest side of the triangle is called the hypotenuse,
∴ 73 is the hypotenuse.

Q10. Check whether the sides form a right angled triangle 35,12 and 34? (Apply Converse of Pythagoras theorem).
Ans:
According to the Pythagoras theorem,
In a right angle triangle square of the hypotenuse is equals to the sum of the square of the other two sides.
In △ABC, let
a = 12cm
b = 34cm
c = 34cm
if a2 + b2 = c2 , then the triangle is right angled triangle.
Let us first find a2 + b2
⇒ 12² + 34²
⇒ 144 + 1156 = 1300
⇒ c² = 35² = 1225
Here 1300 ≠ 1225
Hence, the given sides does not form a right angled triangle.

Q11. Check whether the sides form a right angled triangle 65,72 and 97? (Apply Converse of Pythagoras theorem).
Ans: 
According to the Pythagoras theorem-In a right angle triangle square of the hypotenuse is equals to the sum of the square of the other two sides.
⇒ a² + b² = c²
∴ 65² + 72² = 97²
⇒ 4225 + 5184 = 9409
⇒ 97² = 9408
∴9409 = 9409
Hence, the given sides  form a right angled triangle.

Q12. In a right angled △ABC, ∠B = 90^o , AC = 17cm and AB = 8cm, find BC.
Ans:
Given, in △ABC, ∠B = 90o
∴ AC² = AB² + BC²  [∵ Pythagoras theorem]
⇒ BC² = AC² - AB²
BC² = 17² - 8²
BC² = 289 - 64 = 225
∴ BC = √225
= 15cm

Q13. The sides of a right angled triangle containing the right angle are 5cm and 12cm, find its hypotenuse.
Ans:
Let AB and BC be the sides of the triangle containing right angle and AC be the hypotenuse as shown in the above figure:
It is given that AB=5 cm BC=12 cm
According to the pythagoras theorem
AC² = AB² +BC²
⇒ AC² =5² +12²
⇒ AC² = 25+144
⇒ AC² = 169

⇒ AC= √169
⇒ AC=13
Hence, the hypotenuse AC = 13 cm

Q14. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.
Ans:
Produce AD upto E such that AD=DE
In △ABD and △EDC
AD = DE  [By construction]
BD = CD  [Given]
∠1 = ∠2    [Vertically opposite angles]
∴ △ABD ≅ △EDC   [SAS}
⇒ AB = CE ...........(1)
and ∠BAD = ∠CED
But, ∠BAD = ∠CAD   [AD is bisector of ∠BAC]
∴∠CED = ∠CAD
⇒ AC  = CE...........(2)
From (1) and (2)
AB = AC
Hence, ABC is an isosceles triangle.

Q15. Which of the following will form the sides of a triangle
(i) 23cm, 17cm, 8cm
(ii) 12cm, 10cm, 25cm,
(iii) 9cm, 7cm, 16cm
Ans: 
(i) 23cm, 17cm, 8cm are the given lengths.
Here 23 + 17 > 8, 17 + 8 > 23 and 23 + 8 > 17
∴ 23cm, 17cm, 8cm will form the sides of a triangle.
(ii) 12cm, 10cm, 25cm are the given lengths.
Here 12 +10 is not greater than 25 i.e., [12 + 10 ≯ 25]
∴ 12cm, 10cm, 25cm will not form the sides of a triangle.
(iii) 9cm,7cm,16cm are given lengths 9 + 7 is not greater than 16
i.e., [9 + 7 = 16,9 + 7 ≯ 16]
∴ 9cm, 7cm and 16cm will not be the sides of a triangle.