Mensuration
- The perimeter of a triangle is 30 cm and its area is 30 cm². If the largest side measures 13 cm, what is the length of the smallest side of the triangle ?
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Using Rule 1,
The perimeter of a triangle = 30 cm.
Area = 30 cm²
We know that, 5² + 12² = 13²
∴ The triangle is right angled.Now, Area = 1 × 5 × 12 = 30cm² 2
And, Perimeter = 5 + 12 + 13 = 30
Hence, the smallest side = 5 cmCorrect Option: C
Using Rule 1,
The perimeter of a triangle = 30 cm.
Area = 30 cm²
We know that, 5² + 12² = 13²
∴ The triangle is right angled.Now, Area = 1 × 5 × 12 = 30cm² 2
And, Perimeter = 5 + 12 + 13 = 30
Hence, the smallest side = 5 cm
- The area of a triangle is 216 cm² and its sides are in the ratio 3 : 4 : 5. The perimeter of the triangle is :
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Using Rule 1,
Let the sides be 3x, 4x and 5x respectively.
Here, (3x)² + (4x)² = (5x)²
Hence, the triangle is right angled.∴ 1 × 3x × 4x = 216 2
⇒ 6x² = 216 ⇒ x²216 = 36 6
∴ x = 36 = 6
Perimeter of triangle = (3x + 4x + 5x) cm
= 12x cm = 12 × 6 = 72 cmCorrect Option: D
Using Rule 1,
Let the sides be 3x, 4x and 5x respectively.
Here, (3x)² + (4x)² = (5x)²
Hence, the triangle is right angled.∴ 1 × 3x × 4x = 216 2
⇒ 6x² = 216 ⇒ x²216 = 36 6
∴ x = 36 = 6
Perimeter of triangle = (3x + 4x + 5x) cm
= 12x cm = 12 × 6 = 72 cm
- In a triangular field having sides 30m, 72m and 78m, the length of the altitude to the side measuring 72m is :
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Using Rule 1,
Semi-perimeter,S = 30 + 72 + 78 = 90 2
Area of triangle ABC = √s(s - a)(s - b)(s - c)
= √90(90 - 30)(90 - 72)(90 - 78)
= √90 × 60 × 18 × 12
= 1080 m²
If the altitude AD be h metre, then,1 × Base × height = 1080 2 ⇒ 1 × 72 × h = 1080 2 h = 1080 = 30 metre 36 Correct Option: C
Using Rule 1,
Semi-perimeter,S = 30 + 72 + 78 = 90 2
Area of triangle ABC = √s(s - a)(s - b)(s - c)
= √90(90 - 30)(90 - 72)(90 - 78)
= √90 × 60 × 18 × 12
= 1080 m²
If the altitude AD be h metre, then,1 × Base × height = 1080 2 ⇒ 1 × 72 × h = 1080 2 h = 1080 = 30 metre 36
- If the perimeter of a right-angled isosceles triangle is (4√2 + 4) cm, the length of the hypotenuse is :
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Using Rule 1,
Let ABC be a right-angled isosceles triangle where
AB = BC = x and ∠B = 90°.
∴ AC = √x² + x⊃ = √2x
∴ Perimeter of the triangle ABC = 4√2 + 4
⇒ x + x + √2x = 4√4 + 4
⇒ 2x + 2x = 4√2 + 4
⇒ 2x + 2x = 2.2√2 + 2.2√2
⇒ x = 2√2
∴ Length of the hypotenuse = √2.x = √2 2.√2 = 4 cm.Correct Option: A
Using Rule 1,
Let ABC be a right-angled isosceles triangle where
AB = BC = x and ∠B = 90°.
∴ AC = √x² + x⊃ = √2x
∴ Perimeter of the triangle ABC = 4√2 + 4
⇒ x + x + √2x = 4√4 + 4
⇒ 2x + 2x = 4√2 + 4
⇒ 2x + 2x = 2.2√2 + 2.2√2
⇒ x = 2√2
∴ Length of the hypotenuse = √2.x = √2 2.√2 = 4 cm.
- The sides of a triangle are in the ratio 1/3 : 1/4 : 1/5 and its perimeter is 94cm. The length of the smallest side of the triangle is:
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1 : 1 : 1 3 4 5 or 1 × 60 : 1 × 60 : 1 × 60 3 4 5
or 20 : 15 : 12
∴ 20x + 15x + 12x = 94
[As per question]→ 47x = 94 ⇒ x = 94 = 2 47
∴ The smallest side = 12x
= 12 × 2 = 24 cmCorrect Option: C
1 : 1 : 1 3 4 5 or 1 × 60 : 1 × 60 : 1 × 60 3 4 5
or 20 : 15 : 12
∴ 20x + 15x + 12x = 94
[As per question]→ 47x = 94 ⇒ x = 94 = 2 47
∴ The smallest side = 12x
= 12 × 2 = 24 cm
