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Given base and altitude( height) of a triangle
Where:
a, b, c are the lengths of the sides of the triangle
s is the semi perimeter of the triangle (s = (a + b + c)/2)
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A parallelogram is a four-sided shape with opposite sides that are parallel. The altitude of a parallelogram is the perpendicular distance from a base to the opposite side. The base of a parallelogram is any one of its sides.
The formula for the area of a parallelogram is given by:
A = b \times h
Where:
- *A* is the area of the parallelogram,
- *b* is the length of the base, and
- *h* is the altitude of the parallelogram.
The perimeter of a parallelogram is the sum of the lengths of all its sides. Since opposite sides of a parallelogram are equal in length, we can calculate the perimeter using the following formula:
P = 2 \times (b + s)
Where:
- *P* is the perimeter of the parallelogram,
- *b* is the length of the base, and
- *s* is the length of one of the sides.
The area of a parallelogram can be calculated by multiplying the base by the altitude, while the perimeter can be calculated by summing the lengths of all sides (assuming equal side lengths in a parallelogram). These formulas are crucial in determining the size and dimensions of parallelograms in various mathematical problems.
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Given a parallelogram with base *b = 4 units* and altitude *h = 3 units*, calculate the area and perimeter of the parallelogram.
1. Calculate the area:
A = 4 \times 3 = 12 square units
Therefore, the area of the parallelogram is *12 square units*.
2. Calculate the perimeter (assume all sides are equal):
P = 2 \times (4 + 4) = 16 units
Therefore, the perimeter of the parallelogram is *16 units*.
Question: Given a parallelogram with sides of length 5 cm and 8 cm, and an angle of 60 degrees between these sides, find the area of the parallelogram.
Solution:
\text{Area of a parallelogram} = \text{base} \times \text{height}
= 8 \times 5 \times \sin(60^\circ)
= 8 \times 5 \times \frac{\sqrt{3}}{2}
= 20\sqrt{3} square cm
Question: The diagonals of a parallelogram are 10 cm and 12 cm in length. If the angle between the diagonals is 45 degrees, calculate the area of the parallelogram.
Solution:
\text{Area of a parallelogram} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 \times \sin(45^\circ)
= \frac{1}{2} \times 10 \times 12 \times \sin(45^\circ)
= \frac{1}{2} \times 10 \times 12 \times \frac{\sqrt{2}}{2}
= 60\sqrt{2} square cm
Question: If the base of a parallelogram is 6 cm and the height is 4 cm, find the area of the parallelogram.
Solution:
\text{Area of a parallelogram} = \text{base} \times \text{height}
= 6 \times 4
= 24 square cm
Question: The area of a parallelogram is 72 square units and its base is 9 units long. Find the height of the parallelogram.
Solution:
Let the height of the parallelogram be h. From the given information, we have:
72 = 9 \times h
h = \frac{72}{9}
h = 8 units
Question: If the area of a parallelogram is 48 cm squared and its height is 6 cm, calculate the length of its base.
Solution:
Let the length of the base of the parallelogram be b. From the given information, we have:
48 = b \times 6
b = \frac{48}{6}
b = 8 cm
Question 1: Find the perimeter of a triangle with sides of lengths 4cm, 7cm, and 9cm.
Question 2: Calculate the area of a triangle with base 10cm and height 6cm.
Question 3: Determine the missing angle in a triangle with angles of 45 degrees and 70 degrees.
Question 4: If the sides of a triangle are 8cm, 15cm, and 17cm, is it a right triangle?
Question 5: What is the area of an equilateral triangle with side length 12cm?
Question 6: If a triangle has angles of 30 degrees, 60 degrees, and x degrees, what is the value of x?
Question 7: Determine the height of a triangle with base 9cm and area 27cm^2.
Question 8: Calculate the area of a triangle with side lengths 6cm, 8cm, and 10cm.
Question 9: Find the perimeter of a right triangle with legs of lengths 5cm and 12cm.
Question 10: Determine the missing side length in a triangle with sides of lengths 3cm, 4cm, and x cm.
Question 11: If two sides of a triangle have lengths 7cm and 9cm, and the included angle is 60 degrees, what is the length of the third side?
Question 12: Find the area of a triangle with sides of lengths 5cm, 12cm, and 13cm.
Question 13: Calculate the perimeter of an isosceles triangle with base 10cm and congruent sides of length 8cm.
Question 14: If the angles of a triangle are in the ratio 3:4:5, what is the measure of the largest angle?
Question 15: Determine the missing angle in a triangle with angles of 60 degrees and 80 degrees.
Answer 1: Perimeter = 4 + 7 + 9 = 20 cm
Answer 2: Area = \frac{1}{2} \times 10cm \times 6cm = 30 cm^2
Answer 3: x = 180 - 45 - 70 = 65 degrees
Answer 4: Yes, it is a right triangle.
Answer 5: Area = \frac{\sqrt{3}}{4} \times 12cm^2 = 36\sqrt{3} cm^2
Answer 6: x = 90 degrees
Answer 7: Height = \frac{2 \times Area}{base} = \frac{2 \times 27}{9} = 6 cm
Answer 8: Area = 24 cm^2
Answer 9: Perimeter = 5 + 12 + 13 = 30 cm
Answer 10: x = 5 cm
Answer 11: The length of the third side is 9cm.
Answer 12: Area = 10 cm^2
Answer 13: Perimeter = 8 + 8 + 10 = 26 cm
Answer 14: The largest angle measures 100 degrees.
Answer 15: Missing \, angle = 180 - 60 - 80 = 40 degrees
