Heron S Formula — Andhra Pradesh (SSC) Class 9 Mathematics Solutions (Free)
Free step-by-step Andhra Pradesh (SSC) Class 9 Mathematics solutions for "Heron S Formula" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Andhra Pradesh (SSC) Class 9 Mathematics solutions for "Heron S Formula" — important questions with detailed answers, download PDF f…
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Q1: State Heron's formula for the area of a triangle: For a triangle with sides a, b, c, the semi-perimeter s = (a+b+c)/2, and area = √[s(s-a)(s-b)(s-c)]. Find the area of a triangle with sides 5 cm, 6 cm, and 7 cm.
Step 1: Heron's Formula: For a triangle with sides a, b, and c:
Semi-perimeter s = (a + b + c)/2
Area = √[s(s - a)(s - b)(s - c)]
This formula is used when the height is not given but all three sides are known.
Step 2: Given sides:
a = 5 cm
b = 6 cm
c = 7 cm
Step 3: First, verify that these sides can form a triangle using triangle inequality:
5 + 6 = 11 > 7 ✓
6 + 7 = 13 > 5 ✓
5 + 7 = 12 > 6 ✓
Step 4: Calculate semi-perimeter:
s = (a + b + c)/2
s = (5 + 6 + 7)/2
s = 18/2
s = 9 cm
St…
Q2: Using Heron's formula, find the area of a triangle with sides 10 cm, 10 cm, and 12 cm (isosceles triangle).
Step 1: Given sides of isosceles triangle:
a = 10 cm
b = 10 cm
c = 12 cm
Step 2: Verify triangle inequality:
10 + 10 = 20 > 12 ✓
10 + 12 = 22 > 10 ✓
10 + 12 = 22 > 10 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (10 + 10 + 12)/2
s = 32/2
s = 16 cm
Step 4: Calculate the differences:
s - a = 16 - 10 = 6
s - b = 16 - 10 = 6
s - c = 16 - 12 = 4
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[16 × 6 × 6 × 4]
Area = √[16 × 36 × 4]
Area = √[16 × 144]
A…
Q3: Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm using Heron's formula. Express answer in simplest form.
Step 1: Given sides:
a = 13 cm
b = 14 cm
c = 15 cm
Step 2: Verify triangle inequality:
13 + 14 = 27 > 15 ✓
14 + 15 = 29 > 13 ✓
13 + 15 = 28 > 14 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (13 + 14 + 15)/2
s = 42/2
s = 21 cm
Step 4: Calculate the differences:
s - a = 21 - 13 = 8
s - b = 21 - 14 = 7
s - c = 21 - 15 = 6
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[21 × 8 × 7 × 6]
Step 6: Calculate the product:
21 × 8 = 168
7 × 6 = 42
168 × 42…
Q4: A triangle has sides in the ratio 3:4:5. If its perimeter is 36 cm, find its area using Heron's formula.
Step 1: Given: Sides in ratio 3:4:5, Perimeter = 36 cm
Step 2: Let the sides be 3x, 4x, and 5x (where x is a constant).
Step 3: Use perimeter to find x:
Perimeter = 3x + 4x + 5x = 36
12x = 36
x = 3
Step 4: Find the actual sides:
Side a = 3x = 3 × 3 = 9 cm
Side b = 4x = 4 × 3 = 12 cm
Side c = 5x = 5 × 3 = 15 cm
Step 5: Verify: 9 + 12 + 15 = 36 ✓
Step 6: Note that 9, 12, 15 are multiples of 3, 4, 5, which is a Pythagorean triplet.
Verification: 9² + 12² = 81 + 144 = 225 = 15²
This is a right …
Q5: Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm. Express the answer both in simplified radical form and decimal form.
Step 1: Given sides:
a = 7 cm
b = 8 cm
c = 9 cm
Step 2: Verify triangle inequality:
7 + 8 = 15 > 9 ✓
8 + 9 = 17 > 7 ✓
7 + 9 = 16 > 8 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (7 + 8 + 9)/2
s = 24/2
s = 12 cm
Step 4: Calculate the differences:
s - a = 12 - 7 = 5
s - b = 12 - 8 = 4
s - c = 12 - 9 = 3
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[12 × 5 × 4 × 3]
Area = √[12 × 60]
Area = √720
Step 6: Simplify √720:
720 = 144 × 5 = 12² × 5
√720…
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