Parallelogram. Parallelogram How to find the lower height of a parallelogram
How to determine the height of a parallelogram, knowing some of its other parameters? Such as the area, the lengths of the diagonals and sides, the magnitude of the angles.
You will need
- calculator
Instructions
In problems in geometry, more precisely in planimetry and trigonometry, sometimes it is required to find the height of a parallelogram, based on the given values of the sides, angles, diagonals, etc.
To find the height of a parallelogram, knowing its area and the length of the base, you must use the rule for determining the area of a parallelogram. The area of a parallelogram, as you know, is equal to the product of the height and the length of the base:
S - parallelogram area,
a - the length of the base of the parallelogram,
h is the length of the height lowered to side a, (or to its continuation).
From this we find that the height of the parallelogram will be equal to the area divided by the length of the base:
For example,
given: the area of the parallelogram is 50 sq. cm, the base is 10 cm -
find: the height of the parallelogram.
h = 50/10 = 5 (cm).
Since the height of the parallelogram, the part of the base and the side adjacent to the base form a right-angled triangle, some aspect ratios and angles of right-angled triangles can be used to find the height of the parallelogram.
If the side of the parallelogram adjacent to the height h (DE) is known d (AD) and the angle A (BAD) opposite to the height, then the calculation of the height of the parallelogram must be multiplied by the length of the adjacent side by the sine of the opposite angle:
for example, if d = 10 cm, and the angle A = 30 degrees, then
H = 10 * sin (30?) = 10 * 1/2 = 5 (cm).
If in the conditions of the problem the length of the side of the parallelogram adjacent to the height h (DE) and the length of the part of the base cut off by the height (AE) are specified, then the height of the parallelogram can be found using the Pythagorean theorem:
| AE | ^ 2 + | ED | ^ 2 = | AD | ^ 2, whence we define:
h = | ED | =? (| AD | ^ 2- | AE | ^ 2),
those. the height of the parallelogram is equal to the square root of the difference between the squares of the length of the adjacent side and the part of the base cut off by the height.
For example, if the length of the adjacent side is 5 cm, and the length of the cut-off part of the base is 3 cm, then the length of the height will be:
h =? (5 ^ 2-3 ^ 2) = 4 (cm).
If the length of the diagonal (DВ) of the parallelogram adjacent to the height and the length of the part of the base cut off by the height (BE) are known, then the height of the parallelogram can also be found using the Pythagorean theorem:
| ВE | ^ 2 + | ED | ^ 2 = | ВD | ^ 2, whence we define:
h = | ED | =? (| ВD | ^ 2- | BE | ^ 2),
those. the height of the parallelogram is equal to the square root of the difference between the squares of the length of the adjacent diagonal and the cut-off height (and diagonal) of the part of the base.
For example, if the length of the adjacent side is 5 cm, and the length of the cut-off part of the base is 4 cm, then the length of the height will be:
h =? (5 ^ 2-4 ^ 2) = 3 (cm).
The height of a polygon is a straight line segment perpendicular to one of the sides of the figure, which connects it to the vertex of the opposite corner. There are several such segments in a flat convex figure, and their lengths are not the same if at least one of the sides of the polygon has a different size. Therefore, in problems from the geometry course, it is sometimes required to determine the length of a greater height, for example, a triangle or a parallelogram.
Instructions
Determine which of the heights of the polygon should have the greatest length. In a triangle, this is a segment dropped to the shortest side, so if the dimensions of all three sides are given in the initial conditions, then there is no need to guess.
If, in addition to the length of the shortest of the sides of the triangle (a), in the conditions, the area (S) of the figure is given, the formula for calculating the largest of the heights (H?) Will be quite simple. Double the area and divide the resulting value by the length of the short side - this will be the desired height: H? = 2 * S / a.
Without knowing the area, but having the lengths of all sides of the triangle (a, b and c), you can also find the longest of its heights, but there will be much more mathematical operations. Start by calculating an auxiliary quantity - the semi-perimeter (p). To do this, add the lengths of all sides and divide the result in half: p = (a + b + c) / 2.
Multiply the semiperimeter three times by the difference between it and each side: p * (p-a) * (p-b) * (p-c). From the resulting value, extract the square root? (P * (p-a) * (p-b) * (p-c)) and do not be surprised - you used Heron's formula to find the area of a triangle. To determine the length of the greatest height, it remains to replace the area in the formula from the second step with the resulting expression: H? = 2 *? (P * (p-a) * (p-b) * (p-c)) / a.
A large parallelogram height (H?) Is calculated even easier if the area of this figure (S) and the length of its short side (a) are known. Divide the first by the second and you get the desired result: H? = S / a.
If you know the value of the angle (?) At any of the vertices of the parallelogram, as well as the lengths of the sides (a and b) forming this angle, it will not be very difficult to find the largest of the heights. To do this, multiply the value of the long side by the sine of the known angle, and divide the result by the length of the short side: H? = b * sin (?) / a.
In which the opposite sides are parallel. If a parallelogram has all angles right, then such a parallelogram is called a rectangle, and a rectangle with all sides equal is called a square.
All parallelograms have the following properties:
- opposite sides are equal:
AB = CD and BC = DA
- opposite angles are equal:
∠ABC = ∠CDA and ∠ DAB = ∠BCD
- the sum of the angles adjacent to one side is 180 °:
∠ABC + ∠BCD= 180 °
∠BCD + ∠CDA= 180 °
∠CDA + ∠DAB= 180 °
∠DAB + ∠ABC= 180 ° - at the point of intersection, the diagonals are halved:
AO = OC and BO = OD
- each diagonal divides the parallelogram into two equal triangles:
Δ ABC = Δ CDA and Δ ABD = Δ BCD
- the intersection point of the diagonals is the center of symmetry of the parallelogram:
Point O is the center of symmetry.
Height
The underside of a parallelogram is called it basis, and the perpendicular dropped to the base from any point on the opposite side is height.
AD is the base of the parallelogram, h- height.
Height expresses the distance between opposite sides, so the definition of height can also be formulated as follows: parallelogram height is a perpendicular dropped from any point on one side to the opposite side.
Square
To measure the area of a parallelogram, you can represent it as a rectangle. Consider a parallelogram ABCD:
Constructed heights BE and CF form a rectangle EBCF and two triangles: Δ ABE and Δ DCF... Parallelogram ABCD consists of a quadrilateral EBCD and triangle ABE, rectangle EBCF consists of the same quadrilateral and triangle DCF... Triangles ABE and DCF are equal (according to the fourth criterion of equality of right-angled triangles), which means that the areas of a rectangle with a parallelogram are equal, since they are made up of equal parts.
So, a parallelogram can be represented as a rectangle with the same base and height. And since the lengths of the base and the height are multiplied to find the area of the rectangle, it means that to find the area of the parallelogram, you need to do the same:
square ABCD = AD · BE
From this example, we can conclude that the area of a parallelogram is equal to the product of its base by its height... General formula:
S = ah
where S is the area of the parallelogram, a- base, h- height.
Find the diagonal of the parallelogram drawn from the top of the obtuse angle and the angles it makes with the sides of the parallelogram. Using the cosine theorem, you can find the bisectors of a parallelogram across the sides. If you know the value of the angle (α) at any of the vertices of the parallelogram, as well as the lengths of the sides (a and b) forming this angle, it will not be very difficult to find the largest of the heights.
If, in addition to the length of the shortest of the sides of the triangle (a), the conditions give the area (S) of the figure, the formula for calculating the larger of the heights (Hₐ) will be quite simple. Without knowing the area, but having the lengths of all sides of the triangle (a, b and c), you can also find the longest of its heights, but there will be much more mathematical operations. Start by calculating an auxiliary quantity - the semi-perimeter (p). To do this, add the lengths of all sides and divide the result in half: p = (a + b + c) / 2.
From the resulting value, extract the square root √ (p * (p-a) * (p-b) * (p-c)) and do not be surprised - you used Heron's formula to find the area of a triangle. To determine the length of the greatest height, it remains to replace the area in the formula from the second step with the resulting expression: Hₐ = 2 * √ (p * (p-a) * (p-b) * (p-c)) / a.
Note. This is part of the lesson with geometry problems (section parallelogram). See also: Properties and Area of a Parallelogram. Then, knowing one of the angles, depending on what height was given, we subtract it from 180 degrees to find the second. Using the same cosine theorem, you can find the angle between the diagonals in one of the four triangles formed by them, where the sides are half the diagonals and one of the sides of the parallelogram.
We have a lot of people to help you here. Also, my last question was resolved in less than 10 minutes: D Anyway, you can just go in and try to add your question. A parallelogram is a type of quadrangle, and the height is the perpendicular from the vertex to the opposite side.
Multiply the semiperimeter three times by the difference between it and each side: p * (p-a) * (p-b) * (p-c). To do this, multiply the value of the long side by the sine of the known angle, and divide the result by the length of the short side: Hₐ = b * sin (α) / a. The USE results depend not only on the knowledge and skills of the graduate: it is also important to fill in correctly ...
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If you need to solve a geometry problem that is not here, write about it in the forum. You need to learn how to correctly and COMPLETELY formulate the question. It is necessary to write completely the condition of the problem. A triangle is considered to be isosceles, since from the properties of the bisector and the sum of the angles in a triangle it follows that the angles at the base of such a triangle are congruent. Please help me solve one problem.
Therefore, in problems from the geometry course, it is sometimes required to determine the length of a greater height, for example, a triangle or a parallelogram. The perimeter of a parallelogram, knowing the sides, looks like their doubled sum, and the area is the product of the height and the side by which it is lowered.
A parallelogram is a quadrangle with opposite and pairwise parallel sides to each other.
The height of a parallelogram is a line that is perpendicular to one side of the parallelogram and connects that side to the opposite angle.
In order to find out how to find the length of the parallelogram height, let's turn to the formulas. Height is most often indicated by the letter h.
The method of finding the height depends on the values known to us in the task. Let's consider different methods with specific examples.
Example 1
The area (S) and the length of the base (a) are given.
- Formula: h = S / a
Example: The area of a parallelogram is 100 cm 2, the base to which the height is drawn is 20 cm. Find the height.
- h = 100/20 = 5
- Answer: 5 cm
Example 2
The length of the side of the parallelogram adjacent to the height (b) and the angle opposite to the height itself (a) are given.
- Formula: h = b * sin a
Example: Let's denote our parallelogram with letters ABCD, the height BE passes from angle ABC to side AD. The length of the side AB is 20 cm, the angle BAD is 30 degrees. Find the height.
- h = 20 * sin 30 ° = 20 * 0.5 = 10
Answer: 10 cm
Example 3
The length of the parallelogram side adjacent to the height (n) and the length of the side cut off from the base (m) are given.
- h = root of (n 2 - m 2)
Example: in parallelogram ABCD, the height BE runs from angle ABC to side AD. Length AB is 5 cm, length AE is 3 cm. Find the height.
- h = root of (AD 2 - AB 2)
- h = root of (5 2 -3 2) = 4
- Answer: 4 cm
Example 4
The length of the diagonal extending from the same angle as the height (d) and the length of the side cut off from the base (m) are given.
- h = root of (d 2 - m 2)
Example: in parallelogram ABCD, the height BE runs from angle ABC to side AD. Diagonal BD is 5 cm, length ED = 4 cm.
- h = root of (BD 2 - ED 2)
- h = root of (5 2 - 4 2) = 3
- Answer: 3 cm
If in the task it is required to find a large parallelogram height, then it is necessary to calculate the lengths of both heights and select the largest value.
How to determine the height of a parallelogram, knowing some of its other parameters? Such as the area, the lengths of the diagonals and sides, the magnitude of the angles.
You will need
- calculator
Instructions
1. In problems in geometry, or rather in planimetry and trigonometry, it is sometimes necessary to find the height of a parallelogram based on the given values of the sides, angles, diagonals, etc. The area of the parallelogram, as is known, is equal to the product of the height by the length of the base: S = a * h, where: S is the area of the parallelogram, a is the length of the base of the parallelogram, h is the length of the height lowered to side a, (or its continuation). that the height of the parallelogram will be equal to the area divided by the length of the base: h = S / a For example, given: the area of the parallelogram is 50 sq. cm, the base is 10 cm; find: the height of the parallelogram h = 50/10 = 5 (cm ).
2. Because the height of the parallelogram, part of the base and the side adjacent to the base form a right-angled triangle, it is allowed to use some aspect ratios and angles of right-angled triangles to find the height of the parallelogram. A (BAD), then calculating the height of the parallelogram, you need to multiply the length of the adjacent side by the sine of the opposite angle: h = d * sinA, say, if d = 10 cm, and angle A = 30 degrees, then H = 10 * sin (30?) = 10 * 1/2 = 5 (cm).
3. If in the conditions of the problem the length of the side of the parallelogram adjacent to the height h (DE) and the length of the part of the base cut off by the height (AE) are specified, then the height of the parallelogram can be found using the Pythagorean theorem: | AE | ^ 2 + | ED | ^ 2 = | AD | ^ 2, whence we define: h = | ED | =? (| AD | ^ 2- | AE | ^ 2), i.e. the height of the parallelogram is equal to the square root of the difference between the squares of the length of the adjacent side and the cut-off height of the part of the base. Say, if the length of the adjacent side is 5 cm, and the length of the cut-off part of the base is 3 cm, then the length of the height will be: h =? (5 ^ 2- 3 ^ 2) = 4 (cm).
4. If the length of the diagonal (DB) of the parallelogram adjacent to the height and the length of the part of the base (BE) cut off by the height are famous, then the height of the parallelogram can also be found using the Pythagorean theorem: | BE | ^ 2 + | ED | ^ 2 = | BD | ^ 2, whence we define: h = | ED | =? (| ВD | ^ 2- | BE | ^ 2), i.e. the height of the parallelogram is equal to the square root of the difference between the squares of the length of the adjacent diagonal and the cut-off height (and the diagonal) of the part of the base. Say, if the length of the abutting side is 5 cm, and the length of the cut-off part of the base is 4 cm, then the length of the height will be: h =? ( 5 ^ 2-4 ^ 2) = 3 (cm).
The height of a polygon is a straight line segment perpendicular to one of the sides of the figure, the one that connects it to the vertex of the opposite corner. There are several such segments in a flat convex figure, and their lengths are not identical, if at least one of the sides of the polygon has a good size compared to the others. Consequently, in problems from the geometry course, it is sometimes required to determine the length of a greater height, say, a triangle or a parallelogram.
Instructions
1. Determine which of the heights of the polygon should have the largest length. In a triangle, this is a segment lowered to the shortest side, so if the initial conditions give the sizes of all 3 sides, then you don't have to guess.
2. If, in addition to the length of the shortest of the sides of the triangle (a), the conditions give the area (S) of the figure, the formula for calculating the larger of the heights (H?) Will be rather primitive. Double the area and divide the resulting value by the length of the short side - this will be the desired height: H? = 2 * S / a.
3. Without knowing the area, but having the lengths of all sides of the triangle (a, b and c), it is also allowed to find the longest of its heights, however, the mathematical operations will be much more enormous. Start by calculating an auxiliary quantity - the semi-perimeter (p). To do this, add the lengths of all sides and divide the total in half: p = (a + b + c) / 2.
4. Multiply the semiperimeter three times by the difference between it and any of the sides: p * (p-a) * (p-b) * (p-c). From the resulting value, extract the square root? (P * (p-a) * (p-b) * (p-c)) and do not be amazed - you used Heron's formula to find the area of a triangle. To determine the length of the greatest height, it remains to replace the area in the formula from the second step with the resulting expression: H? = 2 *? (P * (p-a) * (p-b) * (p-c)) / a.
5. The huge height of a parallelogram (H?) Is even easier to calculate if the area of this figure (S) and the length of its short side (a) are known. Divide the first by the second and get the required total: H? = S / a.
6. If we know the value of the angle (?) At any of the vertices of the parallelogram, as well as the lengths of the sides (a and b) forming this angle, it will not be difficult to find the largest of the heights either. To do this, multiply the value of the long side by the sine of the famous angle, and divide the total by the length of the short side: H? = b * sin (?) / a.
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