1. Heron's formula is used to find the area of a triangle given:
Answer:
Explanation:
Heron's formula calculates the area of a triangle when all three side lengths are known.
2. If a triangle has sides of length a, b, and c, the semi-perimeter (s) is calculated as:
Answer:
Explanation:
The semi-perimeter is half the total perimeter of the triangle.
3. According to Heron's formula, the area (A) of a triangle with side lengths a, b, and c, and semi-perimeter s is:
Answer:
Explanation:
Heron's formula expresses the area in terms of the semi-perimeter and the side lengths.
4. For a triangle with sides 5 cm, 12 cm, and 13 cm, its semi-perimeter is:
Answer:
Explanation:
Semi-perimeter, s = (5 + 12 + 13) / 2 = 30 / 2 = 15 cm.
5. If the three sides of a triangle are equal in length, the triangle is:
Answer:
Explanation:
A triangle with all three sides of equal length is an equilateral triangle.
6. Heron's formula can be applied to which of the following triangles?
Answer:
Explanation:
Heron's formula is applicable for any triangle as long as the side lengths are known.
7. If one side of a triangle is longer than the sum of the other two sides, the area of the triangle is:
Answer:
Explanation:
If one side is longer than the sum of the other two, the triangle cannot exist, making its area undefined.
8. The expression "s(s – a)(s – b)(s – c)" is often seen in:
Answer:
Explanation:
This expression forms the core of Heron's formula for calculating the area of a triangle.
9. In Heron's formula, if s is the semi-perimeter, then the perimeter of the triangle is:
Answer:
Explanation:
Since s is half the perimeter, the full perimeter would be 2s.
10. If a triangle has an area of zero according to Heron's formula, then:
Answer:
Explanation:
If the area is zero, it implies that the three points are collinear, and thus, it's not a valid triangle.
11. The value of s(s – a)(s – b)(s – c) must be:
Answer:
Explanation:
Since the area of a triangle cannot be negative, the value inside the square root in Heron's formula must be positive.
12. Heron of Alexandria is also known for:
Answer:
Explanation:
Heron is also known for creating the first recorded steam engine, called an aeolipile.
13. For a triangle with sides 6 cm, 8 cm, and 10 cm, its area using Heron's formula is:
Answer:
Explanation:
Using Heron's formula, we find the semi-perimeter to be 12 cm and the area to be 24 cm^2.
14. A triangle with sides a, b, and c can have an area greater than zero if:
Answer:
Explanation:
For a triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side.
15. If a triangle is isosceles with sides a, a, and b, then its area using Heron's formula is:
Answer:
Explanation:
For an isosceles triangle using Heron's formula, the formula simplifies as given in option a.
16. The main advantage of Heron's formula over the ½ × base × height formula is:
Answer:
Explanation:
Heron's formula comes in handy especially when the triangle's height isn't available.
17. Heron's formula can be derived from:
Answer:
Explanation:
Heron's formula can be derived using the Pythagoras' theorem for right triangles.
18. If in Heron's formula, s is the semi-perimeter and A is the area, then for a triangle with sides a, b, and c:
Answer:
Explanation:
Generally, the area of a triangle is smaller than its semi-perimeter.
19. The dimensions that make the value inside the square root of Heron's formula negative, result in:
Answer:
Explanation:
A negative value inside the square root means the triangle doesn't satisfy the triangle inequality theorem and hence, cannot exist.
20. Heron's formula is especially useful for:
Answer:
Explanation:
When working in coordinate geometry, often only the side lengths are known making Heron's formula very useful.